Deformation equivalence of singular moduli spaces of sheaves on K3 surfaces

Let $m$ be a positive integer. For $i=1$ or $2$, let $X_i$ be a projective K3 surface, let $v_i=(r_i,c_i,a_i)$ be a primitive positive Mukai vector, and let $H_i$ be a polarization generic with respect to the Mukai vector $mv_i$. The moduli space of $H_i$-semistable sheaves with Mukai vector $mv_i$ is denoted by $M_{X_i,H_i}(mv_i)$. Deformation-equivalence conjecture. If

$$\dim M_{X_1,H_1}(mv_1)=\dim M_{X_2,H_2}(mv_2),$$

then

$$M_{X_1,H_1}(mv_1)\quad\text{and}\quad M_{X_2,H_2}(mv_2)$$

are deformation equivalent. The conjecture proposes that, for arbitrary non-primitive Mukai vectors, the deformation type of these singular moduli spaces is determined by their dimension. The corresponding result is established in the paper up to birational maps, but removing those birational maps would require a version of Huybrechts's theorem for singular moduli spaces; the general assertion remains open.