The geometric description of the Braverman–Kazhdan–Polishchuk incidence scheme

Let $\mathcal C$ be a smooth complete irreducible curve of genus $g\geq 2$, and let $\mathcal E$ be a stable bundle on $\mathcal C$ of degree $2g-1$. Define $F_{\mathcal E}$ to be the scheme of pairs $(\mathcal L,s)$ with $\mathcal L\in\operatorname{Pic}^0(\mathcal C)$ and $s\in\mathbb P(H^0(\mathcal C,\mathcal L\otimes\mathcal E))$. Braverman–Kazhdan–Polishchuk's conjecture. The scheme $F_{\mathcal E}$ is irreducible, has dimension $g$, and has rational singularities. The source explains that this statement would imply cases of the analytic conjecture above and is known for curves of genus $2$ and $3$; its general status is otherwise open.