Conjecture on moduli spaces of extremal Chern character sheaves on projective three-space

Let $(r,c,d)\in\mathbb{Z}\oplus\mathbb{Z}\oplus\tfrac{1}{2}\mathbb{Z}$, and suppose that there are Gieseker-semistable sheaves $E$ on $\mathbb{P}^3$ with

$$\operatorname{ch}_{\leq 2}(E)=(r,c,d).$$

Write $E(r,c,d)$ for the extremal third Chern character associated with $(r,c,d)$, and let $M(r,c,d,E(r,c,d))$ denote the moduli space of Gieseker-semistable sheaves with this Chern character. The conjecture. The moduli space $M(r,c,d,E(r,c,d))$ is irreducible and smooth along the open locus of stable sheaves. This conjecture extends a conjecture made by Schmidt for rank-two sheaves and is verified in the paper in additional cases; the general assertion is not stated as solved.