Traditional Humphreys conjecture on support varieties of tilting modules

Assume $p>h$. Let $W_{\mathrm{ext}}^\varnothing$ be the set of elements of the extended affine Weyl group that are minimal in their cosets modulo $W$, let $\mathsf T(\lambda)$ be the indecomposable tilting module of highest weight $\lambda$, let $\mathcal N$ be the nilpotent cone, and let $V_{\mathbf G_1}(M)$ be the support variety of a $\mathbf G_1$-module. For $w\in W_{\mathrm{ext}}^\varnothing$, let $C$ be the $G$-orbit in $\mathcal N$ corresponding via Lusztig's bijection to the two-sided cell containing $w$.

Traditional Humphreys conjecture.

$$V_{\mathbf G_1}(\mathsf T(w\bullet0))=\overline C.$$

Humphreys originally presented this as a hypothesis or statement worthy of inquiry. The supplied text does not establish its resolution.