Mirabolic-invariance conjecture for bilinear forms on homogeneous representations

Let $G=\operatorname{GL}_{mn}$, let $D$ be the subgroup used in the paper, and let $\sigma\in\operatorname{Irr}G$ be $m$-homogeneous. Consider bilinear forms on $\sigma\times\sigma^\vee$. Mirabolic-invariance conjecture. Every $D$-invariant bilinear form on $\sigma\times\sigma^\vee$ is $G$-invariant. The case $m=1$ follows from Bernstein's theorem, and the paper notes additional cases obtained by analytic continuation; the general assertion is left without a supplied resolution.