Bayer–Macrì–Toda generalized Bogomolov inequality for tilt-stable objects

Let $X$ be the threefold under consideration, let $H$ be an ample divisor, and let $u_{\alpha,\beta}$ denote the tilt slope on the tilted heart $\operatorname{Coh}^{\beta}(X)$. For an object $E\in\operatorname{Coh}^{\beta}(X)$, write $\operatorname{ch}^{\beta}_i(E)$ for its $\beta$-twisted Chern character.

Bayer–Macrì–Toda conjecture. For any $\nu_{\alpha,\beta}$-stable object $E\in\operatorname{Coh}^{\beta}(X)$ with $\nu_{\alpha,\beta}(E)=0$, the inequality

$$\operatorname{ch}^{\beta}_3(E)\leq \frac{\alpha^2}{6}H^2\cdot\operatorname{ch}^{\beta}_1(E)$$

holds.

This inequality was conjectured as a generalized Bogomolov-type inequality for the construction of Bridgeland stability conditions on arbitrary threefolds. It is refuted: the conjecture fails for the blow-up of $\mathbb{P}^3$ at a single point, as shown by Schmidt.