Perego's conjecture on moduli spaces of stable sheaves on K3 surfaces

Let $S$ be a K3 surface with a Kähler class $\omega$ and a Mukai vector

$$v=(r,\xi,a)\in H^{2*}(S,\mathbb Z),\, v^{2}\geq 0$$

where $\xi\in \mathrm{NS}(S)$, $r>1$ is prime to $\xi$, and $\omega$ is $v$-generic. Let $M_v(S,\omega)$ be the moduli space of $\mu_\omega$-stable coherent sheaves on $S$ with Mukai vector $v$. Perego's conjecture. The moduli space $M_v(S,\omega)$ is a hyperkähler manifold. The conjecture predicts hyperkähler geometry for these moduli spaces; the source presents it as a conjecture raised by Perego, while the paper uses it as motivation and does not state a resolution.