Generalized semiorthogonal decomposition conjecture for moduli spaces of vector bundles

Let $X$ be a smooth projective curve, let $\mathrm{M}(r,L)$ be the moduli space of stable rank-$r$ vector bundles on $X$ with fixed determinant $L$, and let $\operatorname{Jac}(X)$ denote the Jacobian of $X$. Generalized semiorthogonal decomposition conjecture. The category $\mathrm{D}^b(\mathrm{M}(r,L))$ has a semiorthogonal decomposition whose indecomposable components can each be described as the derived category of a product of symmetric products of $X$ and $\operatorname{Jac}(X)$. This is the expected higher-rank analogue of the rank-two decomposition and is supported by motivic decompositions; an explicit rank-three formulation and further evidence are known, but the general decomposition remains open.