Gunningham–Jordan–Vazirani's compact generation conjecture for derived skein categories

Let $G$ be a reductive algebraic group, let $q$ be generic, and let $\Sigma$ be a closed surface. Let $\widehat{\mathcal{S}\mathrm{kCat}_G(\Sigma)}$ be the indicated stable presentable $\infty$-category.

Compact generation conjecture. The category $\widehat{\mathcal{S}\mathrm{kCat}_G(\Sigma)}$ admits a compact generator.

This is the derived version of a categorical finiteness conjecture for skein categories. The source does not state whether it has been resolved; it emphasizes the nontrivial finite tensor-generation issue for $\mathrm{Rep}_q(G)$.