Kählerness conjecture for moduli spaces of stable sheaves on K3 surfaces

Let $S$ be a K3 surface, let $v=(r,\xi,a)\in H^{2*}(S,\mathbb{Z})$ be a Mukai vector, and let $\omega$ be a $v$-generic Kähler class satisfying the hypotheses under which the moduli space $M_{v}^{\mu}(S,\omega)$ of $\mu_{\omega}$-stable sheaves is a compact, connected complex manifold with a holomorphic symplectic form. Kählerness conjecture. The moduli spaces

$$M_{v}^{\mu}(S,\omega)$$

are Kähler manifolds. If so, they are irreducible hyperkähler manifolds of K3$^{[n]}$-type. This is known when $S$ is projective, when $M_{v}^{\mu}(S,\omega)$ is a surface, and when it parametrizes only locally free sheaves; the general case for non-projective K3 surfaces remains open.