Humphreys' support variety conjecture for tilting modules

Let $G$ be the reductive algebraic group, $G_1$ its first Frobenius kernel, $h$ the Coxeter number, and $W_p^+$ the set of minimum-length right coset representatives in $W\backslash W_p$. For $w\in W_p^+$, let $[w]$ be the right cell containing $w$, let $\mathcal{O}_{[w]}\subseteq\mathcal{N}$ be the nilpotent orbit corresponding to $[w]$ under the Lusztig bijection, and let $T(w\cdot 0)$ be the indecomposable tilting module of highest weight $w\cdot 0$. Humphreys' conjecture. If $p\geq h$, then for every $w\in W_p^+$,

$$V_{G_1}(T(w\cdot 0))=\overline{\mathcal{O}_{[w]}}.$$

This conjecture predicts that the support varieties of indecomposable tilting modules realize the Lusztig bijection between right cells and nilpotent orbits. It is attributed to J.E. Humphreys and is presented here as a major problem motivating the computation of these support varieties.