Steinberg tensor-product vanishing conjecture for $\mathscr{X}({\bf G})$

Let ${\bf G}$ be the group under consideration, let $\Bbbk$ be the coefficient field, let ${\bf T}$ be its torus, and let $\operatorname{St}$ denote the Steinberg module. For a character $\theta\in\widehat{\bf T}$, let $E(\theta)_J$ be the simple object of $\mathscr{X}({\bf G})$ indexed by $\theta$ and $J\subset I(\theta)$. Steinberg tensor-product vanishing conjecture. For every nontrivial character $\theta\in\widehat{\bf T}$ and every $J\subset I(\theta)$, one has

$$\operatorname{Hom}_{\Bbbk {\bf G}}(\operatorname{St}\otimes\operatorname{St},E(\theta)_J)=0.$$

This is presented as a simplification of the weight-support vanishing conjecture and concerns the tensor square of the Steinberg module. The source gives no proof or resolution of this formulation.