The orthogonal Kronecker modules conjecture for generic Harder–Narasimhan filtrations

Let $\mathbb{F}_e$ be a Hirzebruch surface and let $H_m$ be an arbitrary polarization. Let $\mathbf{v}\in K(\mathbb{F}_e)$ be a character such that there are $H_{\lceil m\rceil}$-prioritary sheaves of character $\mathbf{v}$. Let $\mathbf{v}_1,\ldots,\mathbf{v}_\ell$ be the characters of the factors in the $H_m$-Harder–Narasimhan filtration of a general sheaf $\mathcal{V}\in \mathcal{P}_F(\mathbf{v})$. Suppose that more than one of the $\mathbf{v}_i$ is not semiexceptional. Orthogonal Kronecker modules conjecture. Then $\ell=2$, and there is a full exceptional collection $\mathcal{E}_1,\mathcal{E}_2,\mathcal{E}_3,\mathcal{E}_4$ such that $\mathbf{v}_1$ is a linear combination of $\operatorname{ch}\mathcal{E}_3$ and $\operatorname{ch}\mathcal{E}_4$, while $\mathbf{v}_2$ is a linear combination of $\operatorname{ch}\mathcal{E}_1$ and $\operatorname{ch}\mathcal{E}_2$. The conjecture identifies pairs of orthogonal Kronecker modules as the only additional source of generic Harder–Narasimhan filtrations beyond the usual semiexceptional factors. An affirmative answer would allow an exact inductive computation of the function $\delta_m^{\mu\text{-s}}(\nu)$; the source provides no evidence that the conjecture has been resolved.