Humphreys' weight-cell support-variety conjecture

Let $G$ be a semisimple simply connected algebraic group, let $W_p^+$ be the dominant right-coset representatives, let $c_{[w]}$ be the weight cell associated with the right cell $[w]$ containing $w\in W_p^+$, and let $T(\lambda)$ be the indecomposable tilting module of highest weight $\lambda$. For $w\in W_p^+$, denote by $\mathcal{O}_{[w]}$ the nilpotent orbit corresponding to $[w]$ under the Lusztig bijection. Humphreys' weight-cell conjecture. If $p\geq h$ and $\lambda\in c_{[w]}\cap X(T)_+$ for some $w\in W_p^+$, then

$$V_{G_1}(T(\lambda))=\overline{\mathcal{O}_{[w]}}.$$

This is equivalent to Humphreys' original formulation because tilting modules whose highest weights lie in the same relevant alcove have the same support variety; the source states that the conjecture remains open for all types and also makes sense when $p<h$.