Hom-finiteness 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 abelian category of $\Bbbk {\bf G}$-modules defined by the stated condition on their ${\bf T}$-weight spaces. Let $M,N\in\mathscr{X}({\bf G})$, and let $L$ be a simple object of $\mathscr{X}({\bf G})$. Hom-finiteness conjecture. One has

$$\dim_{\Bbbk}\operatorname{Hom}_{\Bbbk {\bf G}}(M\otimes N,L)<\infty.$$

Although tensor products of objects of $\mathscr{X}({\bf G})$ need not belong to the category and their simple quotients need not belong to it either, this conjecture predicts finite-dimensional Hom-spaces from such tensor products to every simple object. The source verifies it for the tensor products $\mathbb{M}(\lambda)_J\otimes\mathbb{M}(\mu)$, but gives no general resolution.