On a functional equation appearing in characterization of distributions by the optimality of an estimate

Abstract: Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $\mathbb{T}$. Let $\mu$ be a probability distribution on $X$ such that its characteristic function $\hat\mu(y)$ does not vanish and $\hat\mu(y)$ for some $n \geq 3$ satisfies the equation $$ \prod_{j=1}^{n} \hat\mu(y_j + y) = \prod_{j=1}^{n}\hat\mu(y_j - y), \quad \sum_{j=1}^{n} y_j = 0, \quad y_1,\dots,y_n,y \in Y. $$ Then $\mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2.