Heyde's conjecture for locally compact Abelian groups without a 2-dimensional torus

Let $X$ be a second countable locally compact Abelian group, let $G$ be the subgroup of $X$ generated by all elements of $X$ of order $2$, and let $\alpha$ be a topological automorphism of $X$ satisfying condition $(\mathrm{d1})$. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ and distributions $\mu_1$ and $\mu_2$ with nonvanishing characteristic functions. Define $L_1=\xi_1+\xi_2$ and $L_2=\xi_1+\alpha\xi_2$. Heyde's conjecture. If the conditional distribution of $L_2$ given $L_1$ is symmetric, then

$$\mu_j\in\Gamma(X)*\operatorname{M^1}(G),\qquad j=1,2,$$

if and only if $X$ contains no subgroup topologically isomorphic to the $2$-dimensional torus $\mathbb{T}^2$. This conjecture extends the Heyde-type characterization from groups without a 2-dimensional torus; the preceding result shows the corresponding phenomenon on $\mathbb{T}^2$ under a different automorphism condition, while the general equivalence for second countable locally compact Abelian groups is presented as an open question.