In our recent work, we resolve a computational problem which arose in combinatorial context, under assumption that $\mathsf{EXP} \ne \mathsf{\oplus{}EXP}$, where $\mathsf{\oplus{}EXP}$ is the $\mathsf{EXP}$-version of $\mathsf{\oplus{}P}$. The only paper on $\mathsf{\oplus{}EXP}$ that we found was the Beigel-Buhrman-Fortnow 1998 paper that is cited on Complexity Zoo. We understand that we can take parity versions of $\mathsf{NEXP}$-complete problems (see this question), but perhaps many of them are in fact not complete in $\mathsf{\oplus{}EXP}$.
QUESTION: Are there complexity reasons to believe that $\mathsf{EXP} \ne \mathsf{\oplus{}EXP}$? Are there natural combinatorial problems that are complete in $\mathsf{\oplus{}EXP}$? Are there some references we might be missing?