If you look at the definitions, you realise that you don't need the research for explaining the relationship between the two.
Cognitive dissonance is the direct (emotional, if you will) result of holding two opposing beliefs at the same time:
In the field of psychology, cognitive dissonance is the mental
discomfort (psychological stress) experienced by a person who
simultaneously holds two or more contradictory beliefs, ideas, or
Confirmation bias is the attempt to "twitch" information opposing to your beliefs so as to not generate a conflict in your cognition, so that it doesn't generate the discomfort that we just defined as cognitive dissonance:
the tendency to search for, interpret, favor, and recall information
in a way that confirms one's preexisting beliefs or hypotheses.
To say that confirmation bias's use is to avoid cognitive dissonance is something that derives from the definition of both.We do indeed avoid holding two opposing beliefs at the same time to avoid the psychological distress caused by it. No need to scientifically prove that. Furthermore, saying that cognitive dissonance is one explanation for Cognitive Bias seems to be an understatement of the above.
Easiest simile is, if you define the symbol "1" as representing one item of anything and the symbol "0" as representing no item at all, you do not have to prove that you can represent adding one item to nothing by the symbol "1". It's a working convention you simply accept.