Relative Humphreys conjecture on support varieties

Assume $p>h$. Let $\mathbb X_+$ be the dominant weights, let $w_\lambda$ be the unique element of minimal length in the double coset $Wt_\lambda W$, let $\mathcal N$ be the nilpotent cone, and let $\overline V_{\mathbf G_1}(M)$ denote the relative support variety. For $\lambda\in\mathbb X_+$, let $C$ be the $G$-orbit in $\mathcal N$ corresponding via Lusztig's bijection to the two-sided cell containing $w_\lambda$.

Relative Humphreys conjecture.

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

This is a variant of the traditional Humphreys conjecture for relative support varieties. The supplied text does not state whether it has been resolved.