Categorical Finiteness Conjecture for skein categories

Let $G$) be a reductive group, let $\Sigma$ be a closed compact surface, and let $\mathrm{SkCat}_G(\Sigma)$ denote the $G$-skein category at a generic value of the quantization parameter. Categorical Finiteness Conjecture. The abelian category $\mathrm{SkCat}_G(\Sigma)\textrm{-mod}$ admits a compact projective generator. This conjecture categorifies the finite-dimensionality of skein modules and is known for all closed surfaces when $G=\mathrm{SL}_2$; the general case remains open.