Weight-support vanishing conjecture for tensor products in the category $\mathscr{X}({\bf G})$

Let ${\bf G}$ be the group under consideration, let $\Bbbk$ be the coefficient field, and let $\mathscr{X}({\bf G})$ be the stated category of $\Bbbk {\bf G}$-modules. For a ${\bf T}$-module $V$, write

$$\mathbb{X}_{\bf T}(V)=\{\theta\in\widehat{\bf T}\mid V_\theta\ne0\}$$

for its set of ${\bf T}$-weights. Let $M,N\in\mathscr{X}({\bf G})$, and let $L$ be a simple object of $\mathscr{X}({\bf G})$. Weight-support vanishing conjecture. If

$$\mathbb{X}_{\bf T}(L)\cap\mathbb{X}_{\bf T}(M\otimes N)=\varnothing,$$

then

$$\operatorname{Hom}_{\Bbbk {\bf G}}(M\otimes N,L)=0.$$

This conjecture asserts that a simple target cannot receive a nonzero morphism from the tensor product when their ${\bf T}$-weight supports are disjoint. The source verifies it for $\mathbb{M}(\lambda)_J\otimes\mathbb{M}(\mu)$, but does not resolve it for arbitrary objects of $\mathscr{X}({\bf G})$.