The limiting Bogomolov–Gieseker conjecture for tilt-stable objects

Let $X$ be a smooth projective threefold. For an object $E\in D^b(X)$, let $\overline{\beta}(E)$ be the limiting value of the parameter $\beta$ associated with the kernel of the reduced central charge. Suppose there is an open neighborhood $U\subset\mathbb R^2$ of $(0,\overline{\beta}(E))$ such that, for every $(\alpha,\beta)\in U$ with $\alpha>0$, either $E$ or $E[1]$ is a $\nu_{\alpha,\beta}$-stable object of $\operatorname{Coh}^\beta(X)$. The limiting Bogomolov–Gieseker conjecture. Under these assumptions,

$$\operatorname{ch}_3^{\overline{\beta}(E)}(E)\leq 0.$$

This is a limiting case of the generalized Bogomolov–Gieseker conjecture and is intended to control objects stable near the boundary $\alpha=0$. Its validity is not established in general.