Scheme-theoretic traditional Humphreys conjecture

Assume $p>h$. Let $W_{\mathrm{ext}}^\varnothing$ be the set of minimal coset representatives, let $\mathsf T(\lambda)$ be the indecomposable tilting module, let $\mathcal N$ be the nilpotent cone, and let $\overline C$ be the reduced closed subscheme associated with the orbit closure of the orbit corresponding to the two-sided cell containing $w$.

Scheme-theoretic traditional Humphreys conjecture. For every $w\in W_{\mathrm{ext}}^\varnothing$, the coherent sheaf

$$\operatorname{Ext}^\bullet_{\mathsf{Rep}(\mathbf G_1)}(\mathsf T(w\bullet0),\mathsf T(w\bullet0))$$

is scheme-theoretically supported on $\overline C$.

This is described as a priori stronger than the set-theoretic traditional Humphreys conjecture. The supplied text does not state whether it has been resolved.