Le Potier's strange duality conjecture for Hilbert squares of K3 surfaces

Let $S$ be a polarized $K3$ surface of genus g e#### with Mukai vectors $v=(1,0,-1)$ and $w=(2,L,2)$, and let ${\mathcal M}(v)$ and ${\mathcal M}(w)$ be the corresponding moduli spaces of sheaves. The determinant line-bundle isometries identify ${\mathcal M}(v)$ with $S^{[2]}$, identify $L_2-2\delta$ with $\theta_v(w)$, and define $H=\theta_w(v)$ on ${\mathcal M}(w)$. The divisor defined by the jump locus induces a natural map

$$H^0(S^{[2]},L_2-2\delta)^\vee\longrightarrow H^0({\mathcal M}(w),H).$$

Strange duality conjecture. For any $g\geq6$, this map is an isomorphism.

This is Le Potier's strange duality statement for the pair $(S^{[2]},L_2-2\delta)$. The source says that it generalizes O'Grady's result for $6\leq g\leq8$; the claim is presented as proved in the subsection, but the supplied parser status is unknown, so its resolution should be checked.