A refinement of the strange duality conjecture on K3 surfaces

Let $v,w$ be primitive Mukai vectors of positive rank on a polarized K3 surface with $χ(v\otimes w)=0$. Marian and Oprea conjectured that the strange duality map $H^0(M(v),Θ_w)^\vee\to H^0(M(w),Θ_v)$ is either zero or an isomorphism. We give three families of counterexamples to this conjecture. In e