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

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 $hatmu(y)$ does not vanish and $hatmu(y)$ for some $n geq 3$ satisfies the equation $$ prod_{j=1}^{n} hatmu(y_j + y) = prod_{j=1}^{n}hatmu(y_j - y), quad sum_{j=1}^{n} y_j = 0, quad y_1,dots,y_n,y in Y. $$ Then $mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2.