Existence of coarse moduli spaces for semistable derived objects

Let $X$ be a nonsingular projective variety, let $\sigma\in\operatorname{Stab}(X)$ be a stability condition, and let $\alpha\in\mathcal{N}(X)$ be a numerical equivalence class. Moduli-space conjecture. There exists a coarse moduli space $\mathcal{M}_\sigma(\alpha)$ of objects in $D^b\operatorname{Coh}(X)$ of type $\alpha$ that are semistable with respect to $\sigma$. The existence of such spaces would provide a general moduli theory for semistable objects in derived categories; the statement is posed as a hope and is not asserted as a theorem in the source.