The quantum–modular support-variety correspondence conjecture

Let $G$ be a semisimple simply connected algebraic group over a field $k$ of characteristic $p>h$, let $G_1$ be its Frobenius kernel, and let $\mathbb{U}_{\zeta}(\mathfrak{g})$ be the corresponding quantum group for a primitive $p$th root of unity $\zeta\in\mathbb{C}$. Let $V_{\mathbb{C}}\subseteq\mathcal{N}_{\mathbb{C}}$ be a $G_{\mathbb{C}}$-stable closed subvariety, and let $V_k\subseteq\mathcal{N}_k$ be the unique corresponding $G_k$-stable closed subvariety. For a tilting $G_k$-module $M$, write $M_{\zeta}$ for its quantum-group specialization. Quantum–modular support-variety correspondence conjecture. For any tilting module $M$ for $G_k$,

$$V_{\mathfrak{u}_{\zeta}(\mathfrak{g})}(M_{\zeta})=V_{\mathbb{C}}\quad\text{if and only if}\quad V_{G_1}(M)=V_k.$$

The conjecture seeks to realize the correspondence between stable closed subvarieties of the complex and modular nilpotent cones through support varieties. The source does not provide a resolution status, so it is recorded as open.