Searching the first three pages of "personality gender egalitarianism" on Google Scholar, I found five studies on this topic. All of them support the view finding of increased gender differences in more egalitarian countries, and I didn't find any that were against. [Edit: added a Falk 2018 as noted in a comment.]
Most of the studies analyze the Big Five personality traits. Schmitt 2017 also claims increased differences in "Dark Triad traits, self‐esteem, subjective well‐being, depression and values", and Falk 2018 finds increased differences in "willingness to take risks, patience, altruism, positive and negative reciprocity, and trust". One study, Lippa 2010 argues that the differences in the Big Five are quite small, whereas "gender differences on the people–things dimension of interests are ‘very large’". The study authors agree that this shows some inadequacies in social role theory and is more consistent, attributional, and social comparison theories.
Costa Jr., Paul T.,Terracciano, Antonio,McCrae, Robert R.
Journal of Personality and Social Psychology, Vol 81(2), Aug 2001, 322-331. link
Secondary analyses of Revised NEO Personality inventory data from 26 cultures (N =23,031) suggest that gender differences are small relative to individual variation within genders; differences are replicated across cultures for both college-age and adult samples, and differences are broadly consistent with gender stereotypes: Women reported themselves to be higher in Neuroticism, Agreeableness, Warmth, and Openness to Feelings, whereas men were higher in Assertiveness and Openness to Ideas. Contrary to predictions from evolutionary theory, the magnitude of gender differences varied across cultures. Contrary to predictions from the social role model, gender differences were most pronounced in European and American cultures in which traditional sex roles are minimized. Possible explanations for this surprising finding are discussed, including the attribution of masculine and feminine behaviors to roles rather than traits in traditional cultures.
Schmitt, David P.,Realo, Anu,Voracek, Martin,Allik, Jüri
Journal of Personality and Social Psychology, Vol 94(1), Jan 2008, 168-182. link:
Previous research suggested that sex differences in personality traits are larger in prosperous, healthy, and egalitarian cultures in which women have more opportunities equal with those of men. In this article, the authors report cross-cultural findings in which this unintuitive result was replicated across samples from 55 nations (N = 17,637). On responses to the Big Five Inventory, women reported higher levels of neuroticism, extraversion, agreeableness, and conscientiousness than did men across most nations. These findings converge with previous studies in which different Big Five measures and more limited samples of nations were used. Overall, higher levels of human development--including long and healthy life, equal access to knowledge and education, and economic wealth--were the main nation-level predictors of larger sex differences in personality. Changes in men's personality traits appeared to be the primary cause of sex difference variation across cultures. …
Lippa, R. A. (2010), Gender Differences in Personality and Interests: When, Where, and Why?. Social and Personality Psychology Compass, 4: 1098-1110. doi:10.1111/j.1751-9004.2010.00320.x. link
How big are gender differences in personality and interests, and how stable are these differences across cultures and over time? To answer these questions, I summarize data from two meta‐analyses and three cross‐cultural studies on gender differences in personality and interests. Results show that gender differences in Big Five personality traits are ‘small’ to ‘moderate,’ with the largest differences occurring for agreeableness and neuroticism (respective ds = 0.40 and 0.34; women higher than men). In contrast, gender differences on the people–things dimension of interests are ‘very large’ (d = 1.18), with women more people‐oriented and less thing‐oriented than men. Gender differences in personality tend to be larger in gender‐egalitarian societies than in gender‐inegalitarian societies, a finding that contradicts social role theory but is consistent with evolutionary, attributional, and social comparison theories. In contrast, gender differences in interests appear to be consistent across cultures and over time, a finding that suggests possible biologic influences.
Schmitt, D. P., Long, A. E., McPhearson, A. , O'Brien, K. , Remmert, B. and Shah, S. H. (2017), Personality and gender differences in global perspective. Int J Psychol, 52: 45-56. doi:10.1002/ijop.12265. link
… Several large cross‐cultural studies have generated sufficient data for evaluating these global personality predictions. Empirically, evidence suggests gender differences in most aspects of personality—Big Five traits, Dark Triad traits, self‐esteem, subjective well‐being, depression and values—are conspicuously larger in cultures with more egalitarian gender roles, gender socialization and sociopolitical gender equity. Similar patterns are evident when examining objectively measured attributes such as tested cognitive abilities and physical traits such as height and blood pressure. Social role theory appears inadequate for explaining some of the observed cultural variations in men's and women's personalities. Evolutionary theories regarding ecologically‐evoked gender differences are described that may prove more useful in explaining global variation in human personality.
Falk, Armin and Hermle, Johannes (2018), Relationship of gender differences in preferences to economic development and gender equality. Science. doi:10.1126/science.aas9899. link
[According to the resource hypothesis,] greater availability of material and social resources removes the gender-neutral goal of subsistence, which creates the scope for gender-specific ambitions and desires. In addition, more gender-equal access to those resources may allow women and men to express preferences independently from each other. As a consequence, one would expect gender differences in preferences to be positively associated with higher levels of economic development and gender equality.
We tested these competing predictions using data on experimentally validated measures of willingness to take risks, patience, altruism, positive and negative reciprocity, and trust for 80,000 individuals in 76 representative country samples. So that the data would be geographically representative, the dataset was chosen so as to include all continents and a broad range of cultures and economic development levels. In total, the data represent about 90% of both the world population and global income.
The data revealed substantial cross-country variation in gender differences in preferences. Gender differences were found to be strongly positively associated with economic development as well as gender equality. These relationships held for each preference separately as well as for a summary index of differences in all preferences jointly. Quantitatively, this summary index exhibited correlations of 0.67 (P < 0.0001) with log GDP per capita and 0.56 (P < 0.0001) with a Gender Equality Index (a joint measure of four indices of gender equality), respectively. To isolate the separate impacts of economic development and gender equality, we conducted a conditional analysis, finding a quantitatively large and statistically significant association between gender differences and log GDP per capita conditional on the Gender Equality Index, and vice versa. These findings remained robust in several validation tests, such as accounting for potential culture-specific survey response behavior, aggregation bias, and nonlinear relationships.
Mac Giolla, E. and Kajonius, P. J. (2019), Sex differences in personality are larger in gender equal countries: Replicating and extending a surprising finding. Int J Psychol. doi:10.1002/ijop.12529. link
Sex differences in personality have been shown to be larger in more gender equal countries. We advance this research by using an extensive personality measure, the IPIP‐NEO‐120, with large country samples (N > 1000), from 22 countries. Furthermore, to capture the multidimensionality of personality we measure sex differences with a multivariate effect size (Mahalanobis distance D). Results indicate that past research, using univariate measures of effect size, have underestimated the size of between‐country sex differences in personality. Confirming past research, there was a strong correlation (r = .69) between a country's sex differences in personality and their Gender Equality Index. Additional analyses showed that women typically score higher than men on all five trait factors (Neuroticism, Extraversion, Openness, Agreeableness and Conscientiousness), and that these relative differences are larger in more gender equal countries.