Beauville's symplectic strange-duality conjecture

Let $\mathcal M_{Sp_r}$ be the moduli space of semistable pairs $(E,\phi)$, where $E$ has rank $2r$, trivial determinant, and a nondegenerate alternating form $\phi:\Lambda^{2}E\to\mathcal O_X$. Let $\widehat{\mathcal M}_{Sp_k}$ be the analogous moduli space with determinant $K_X^k$ and form $\psi:\Lambda^{2}F\to K_X$. Their determinant bundles are denoted by $\mathcal L_r$ and $\widehat{\mathcal L}_k$. The tensor-product construction gives

$$\mathsf {D}: H^{0}(\mathcal M_{Sp_r}, \mathcal L_r^{k})^{\vee}\to H^{0}(\widehat{\mathcal M}_{Sp_{k}},\widehat{\mathcal L}_k^{r}).$$

Beauville's symplectic strange-duality conjecture. The morphism $\mathsf {D}$ is an isomorphism. Beauville established equality of the dimensions of the two spaces of sections, but the isomorphism itself remains the conjectural part described in the paper.