Cooper's support-variety conjecture for tilting modules

Let $G$ be a semisimple simply connected algebraic group over a field of characteristic $p$, with Frobenius kernel $G_1$, weight lattice $X(T)$, and nilpotent cone $\mathcal{N}=\mathcal{N}(G)$. For $\lambda\in X(T)_+$, define

$$\Gamma_{\lambda}=\{\alpha\in\Phi^+\mid\langle\lambda+\rho,\alpha\rangle\geq p\},$$

and let $s(\lambda)$ be the supremum of the partitions associated with positive subroot systems contained in $\Gamma_{\lambda}$; write $s(\lambda)^t$ for its transpose and $\mathcal{O}_{s(\lambda)^t}$ for the corresponding nilpotent orbit. Let $T(\lambda)$ be the indecomposable tilting module of highest weight $\lambda$. Cooper's conjecture. For every $\lambda\in X(T)_+$,

$$V_{G_1}(T(\lambda))=\overline{\mathcal{O}_{s(\lambda)^t}}.$$

The source describes this as a more general conjecture with no assumption on $p$ and notes that it was verified for $p=2$ by Cooper; its general status is otherwise open.