Lefschetz decomposition conjecture for moduli spaces of rank-two bundles

Let $C$ be a smooth projective curve of genus $g$, let ${\mathcal L}$ be the fixed determinant, and let ${\moduli_C(2,\mathcal L)}$ be the moduli space of rank-two bundles with determinant ${\mathcal L}$. Let $\Theta$ be the polarization used in the paper. Lefschetz decomposition conjecture. There exists a minimal Lefschetz decomposition of $D^b\operatorname{Coh}({\moduli_C(2,\mathcal L)})$ of length $2$ with respect to $\Theta$, whose first block has a semiorthogonal decomposition

$${\mathcal A}_0=\langle D^b\operatorname{Coh}(\mathrm{pt}),D^b\operatorname{Coh}(C),D^b\operatorname{Coh}(\operatorname{Sym}^2 C),\ldots,D^b\operatorname{Coh}(\operatorname{Sym}^{g-1}C)\rangle,$$

so that the residual category is $D^b\operatorname{Coh}(\operatorname{Sym}^{g-1}C)$. This refines the proposed geometrically meaningful semiorthogonal decomposition; the source gives no proof or resolution.