Narasimhan's semiorthogonal decomposition conjecture for the rank-two moduli space

Let $X$ be a smooth projective curve of genus $g\geq 2$, let $L\in\operatorname{Pic}^{d}(X)$ with $(2,d)=1$ and $0<d<2$, and let $\mathrm{M}(2,L)$ be the moduli space of stable rank-two vector bundles on $X$ with determinant $L$. For each $k$, write $X_k=X^k/S_k$ for the $k$-th symmetric product of $X$. Narasimhan's conjecture. The category $\mathrm{D}^b(\mathrm{M}(2,L))$ has a semiorthogonal decomposition

$$\mathrm{D}^b(\mathrm{M}(2,L))=\left\langle\{\mathrm{D}^b(X_k),\mathrm{D}^b(X_k)\}_{0\leq k\leq g-2},\mathrm{D}^b(X_{g-1})\right\rangle.$$

This conjecture was motivated by the embedding results for the derived categories of the symmetric products and by compatible motivic decompositions; those embedding results are known in several cases, but whether the indicated components span the whole derived category remains open.