Bridgeland's moduli-space birationality conjecture for K3 surfaces

Let $X$ be a smooth projective K3 surface, let $\sigma$ be a stability condition on $X$, and let $v$ be a numerical class. A coarse moduli space $M_{\sigma}(v)$ of $\sigma$-semistable complexes with class $v$ should exist. As the stability condition varies, the resulting coarse moduli spaces should be related by birational maps.

Bridgeland's conjecture. Given a stability condition $\sigma$ on a K3 surface and a numerical class $v$, there exists a coarse moduli space $M_{\sigma}(v)$ of $\sigma$-semistable complexes with class $v$, and changing the stability condition produces birational maps between the coarse moduli spaces.

This conjecture concerns the variation of moduli spaces of semistable complexes on K3 surfaces and predicts that wall crossing in the space of stability conditions is reflected by birational geometry. The supplied context does not state whether the conjecture has been resolved.