Uniqueness conjecture for mirabolic-invariant bilinear forms

Let $G=\operatorname{GL}_{mn}$, let $D$ be the subgroup used in the paper, and let $\sigma,\sigma'\in\operatorname{Irr}G$ be $m$-homogeneous. Consider bilinear forms on $\sigma\times\sigma'$. Uniqueness conjecture. For any $m$-homogeneous $\sigma,\sigma'\in\operatorname{Irr}G$, there is a unique up to scalar $D$-invariant bilinear form on $\sigma\times\sigma'$. This is proposed as a stronger statement than the preceding invariance conjecture; no resolution is supplied in the given text.