Strange duality for Mukai moduli spaces on K3 and abelian surfaces

Let $X$ be a $K3$ or abelian surface. Let $v$ and $w$ be primitive, positive Mukai vectors satisfying $\chi(v\otimes w)=0$ and

$$c_{1}(v\otimes w)\cdot H>0.$$

For a $K3$ surface, let $(\mathcal M_v,\mathcal M_w)=({\mathfrak M}_v,{\mathfrak M}_w)$; for an abelian surface, let it be one of $(K_v,\mathfrak M_w)$, $(K_w,\mathfrak M_v)$, $({\mathfrak M}^{+}_v,{\mathfrak M}^{+}_w)$, or $({\mathfrak M}^{-}_v,{\mathfrak M}^{-}_w)$. The duality morphism is

$$\mathsf {D}:H^{0}(\mathcal M_{v}, \Theta_{w})^{\vee} \to H^{0}(\mathcal M_{w}, \Theta_{v}).$$

Mukai strange-duality conjecture. The duality morphism $\mathsf {D}$ is either an isomorphism or zero. Numerical symmetry and vanishing results motivate this dichotomy, while the paper does not determine which alternative occurs in general.