Alright, so I'll give this question a go... Admittedly, I'm not really familiar with theories of emotion lateralization, but let's look at some meta-analyses for insight!!!
As a quick theoretical note, a single study cannot confirm emotion-related lateralization. The representation of emotions or affect in the brain can change from context to context, person to person, and over time. This is why we need to rely on meta-analysis (in my opinion)!
This is a meta-analysis of positive and negative affect in the brain. Overall, the meta-analysis shows large correspondence between the brain regions that are involved in positive vs. negative affect (in favor of a valence-general "workspace"). Moreover, affect seems to be represented bilaterally in most regions as well as right temporal/occipital cortex (see image below).
These areas represent the "salience network," which is involved in self-relevant attention (e.g., to internal bodily sensations and "feelings").
However, the above image shows that the left amygdala and the left insula are more often involved in negative affect than positive affect. No regions seem to show preference for positive affect over negative affect.
This is a meta-analysis using very, very fancy statistical techniques to differentiate among discrete emotion categories (e.g., fear, disgust, anger, etc.).
The above image (B) shows activation intensity in cortex, amygdala, basal ganglia, and thalamus for each emotion. You can see that some regions are more strongly active for certain emotions in a lateralized manner (R and L mean Right and Left Hemisphere).
The most significant lateralization was for happiness, which showed "left-hemisphere dominance in the hippocampus" (p. 16).
While these meta-analyses provide interesting clues about emotion lateralization, I'm going to go ahead and say that we don't really know whether emotion is lateralized. Moreover, the idea of "emotional processing" is incredibly vague and is likely constituted by more basic psychological processes (e.g., interoception and exteroception).
The theory about sympathetic/parasympathetic afferent streams to the right vs. left insula seems plausible (outlined in Wikipedia; Craig), especially given the importance of interoception to affect and emotions (Barrett & Simmons, 2015).
However, overall, it's just not clear.