Maximal third Chern character conjecture for stable sheaf moduli spaces

Let $M(r,c,d,e)$ denote the moduli space of Gieseker-semistable sheaves on ${\mathbb{P}}^3$ with Chern character $(r,c,d,e)$. Let $r \in {\mathbb{Z}}_{\geq 0}$, $c \in {\mathbb{Z}}$, $d \in \tfrac{1}{2}{\mathbb{Z}}$, and $e \in \tfrac{1}{6}{\mathbb{Z}}$ be such that

$$M(r,c,d,e)\ne\emptyset,$$

but $M(r,c,d,e')=\emptyset$ for every $e'>e$. Assume that all Gieseker-semistable sheaves with Chern character $(r,c,d,e)$ are Gieseker-stable. Maximal third Chern character conjecture. Then $M(r,c,d,e)$ is smooth and irreducible. This generalizes the observed smoothness and irreducibility of moduli spaces at maximal third Chern character, including the rank-one and rank-two examples discussed in the paper. The claim is presented as a conjecture; no resolution is supplied in the source.