Aizenbud-Gourevitch-Sayag conjecture on twisted parabolic induction of Gelfand pairs

Let $G$ be a reductive group, and let $P=LU$ be a Levi decomposition of a parabolic subgroup of $G$, with $L$ and $U$ the corresponding Levi and unipotent subgroups. Let $H_0$ be a subgroup of $L$ and let $\psi$ be a character of $H_0U$ whose restriction to $U$ is generic. A triple $(G,H,\psi)$ is a twisted Gelfand pair when $\operatorname{Ind}_{H}^{G}\psi$ is multiplicity-free. Aizenbud-Gourevitch-Sayag conjecture. If $(L,H_0,\psi|_{H_0})$ is a twisted Gelfand pair, then $(G,H_0U,\psi)$ is a twisted Gelfand pair. This conjecture predicts that twisted Gelfand-pair properties are preserved under the indicated parabolic induction, generalizing the analogous construction for spherical pairs. Its resolution is not specified in the source.