Support-variety analogue of the Borel–Bott–Weil theorem

Let $G$ be semisimple and simply connected, with $p$ good, and let $\lambda\in X(T)$ be arbitrary. Suppose $w\in W$ is such that $w\cdot\lambda\in X(T)_+$ and that $H^i(\lambda)\neq 0$ for some $i$. Support-variety conjecture. Then

$$V_{G_1}(H^i(\lambda))=V_{G_1}(H^0(w\cdot\lambda)).$$

This conjecture extends the known determination of support varieties for induced modules $H^0(\lambda)$ to higher cohomology groups, serving as an analogue of the Borel–Bott–Weil theorem for support varieties. Its resolution status is not specified in the source.