The finite-type extension conjecture for coherent Satake categories

Let $C$ be a finite-type Cartan matrix, $G$ the associated simply-connected simple algebraic group, and $G_{Ad}$ its adjoint form. Let $\mathcal{P}_{coh}^{G(\mathcal{O}) \rtimes \mathbb{G}_m}(\mathrm{Gr}_{G_{Ad}})$ denote the category of perverse coherent sheaves in question, and let $\omega_i^\vee$ be a fundamental coweight. Define

$$\widetilde{B}_C = \begin{pmatrix} C^T - C & -C^T \\\\ C & 0 \end{pmatrix}.$$

Finite-type extension conjecture. The category $\mathcal{P}_{coh}^{G(\mathcal{O}) \rtimes \mathbb{G}_m}(\mathrm{Gr}_{G_{Ad}})$ is a monoidal categorification of the quantum cluster algebra $\mathcal{A}^{loc}_{(L_C,\widetilde{B}_C)}$, where $L_C$ is a suitable coefficient matrix; the elements of the initial monoidal cluster are the sheaves $\mathcal{P}_{\omega_i^\vee\!,0}$ and $\mathcal{P}_{\omega_i^\vee\!,\omega_i}$. This would extend the established type-$A$ picture to all finite types and connect simple perverse coherent sheaves with the cluster-theoretic structures expected for line operators. The conjecture remains unresolved in the supplied source.