Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of https://mathoverflow.net/questions/357655/a-sum-of-two-binomial-random-variables :
Let $X$ and $Y$ be $G$-valued independent random variables with $\mu$-density $f_X$ resp. $f_Y$. Let $f_{X \pm Y}$ be the $\mu$-density of $X \pm Y$. Is it always true that
$$\int f_{X-Y}^2 d\mu = \int f_{X+Y}^2 d\mu \ ?$$
This holds true for $G = \mathbb{Z}$.