Let $Y:=Q_1\cap Q_2\subset\mathbb{P}^{n-1}$ be smooth complete intersection of two quadrics. If $n$ is even, then it admits a semi-orthogonal decomposition: $$D^b(Y)=\langle D^b(C),\mathcal{O}_Y,\ldots,\mathcal{O}_Y(n-5)\rangle.$$ Here $C$ is the hyperelliptic curve, which is a double cover $\mathbb{P}^1$ with m critical values. If $n=6$, then $Y$ is a degree $4$ del Pezzo threefold. In this case, $Y$ can be reconstructed as moduli space of stable vector bundle of rank two over $C$(which is a genus two curve, double cover of $\mathbb{P}^1$ at 6 points) with fixed determinant of odd degree. Equivalently, $Y$ can be reconstructed as Brill-Noether locus
$$Y\cong\{F\in M_C(2,1)\, \colon \ \mathrm{dim}_{\mathbb{C}}\mathrm{Hom}(F,\mathcal{R}[1]) \geq 5\}\cong\{F\in M_C(2,1)\, \colon\mathrm{dim}_{\mathbb{C}}\mathrm{Hom}(F,\mathcal{R}) \geq 1\}$$, where $\mathcal{R}$ is a second Raynaud bundle which is isomorphic to $i^!\mathcal{O}_Y[-1]$, where $i:D^b(C)\simeq\mathcal{K}u(Y)\hookrightarrow D^b(Y)$, is the inclusion functor.
My question is for higher dimensional intersection of two quadrics, say $n\geq 8$ and $n$ is even, whether it is known that $Y$ is isomorphic to moduli space of stable vector bundle over such a curve $C$ or its sublocus.
