Ajjanagadde's entropy conjecture for sums from Sidon sets

Let $A$ be a Sidon set in an ambient abelian group, and let $X$ and $Y$ be independent random variables taking values in $A$. Let $H$ denote Shannon entropy. Ajjanagadde's conjecture. The entropy $H(X+Y)$ achieves its maximum when the distributions of $X$ and $Y$ are uniform on $A$. The conjecture concerns addition over sets with minimal additive structure; the source attributes it to Ganesh Ajjanagadde in private communication and gives no evidence of a resolution.