Bayer–Macrì–Toda stability conjecture for threefolds

Let $X$ be a smooth projective threefold, let $H$ be an ample divisor, and let $\betain\mathbb{R}$ and $\alpha>0$. Define

$$\operatorname{ch}^{\beta}(E)=e^{-\beta H}\operatorname{ch}(E)$$

and let $\mathcal{A}_{H}^{\alpha,\beta}$ be the twice-tilted heart with central charge

$$Z^{3}_{\alpha,\beta}(E)=\left(-\operatorname{ch}_{3}^{\beta}(E)+\frac{\alpha^{2}}{2}H^{2}\operatorname{ch}_{1}^{\beta}(E)\right)+i\left(\alpha H\operatorname{ch}_{2}^{\beta}(E)-\frac{\alpha^{3}}{6}H^{3}\operatorname{ch}_{0}^{\beta}(E)\right).$$

Bayer–Macrì–Toda stability conjecture. The pair $(\mathcal{A}_{H}^{\alpha,\beta},Z^{3}_{\alpha,\beta})$ is a stability condition on $D^{b}(X)$. This conjecture is equivalent to the corresponding generalized Bogomolov–Gieseker inequality for tilt-semistable objects on smooth projective threefolds; it is known in several cases but remains open in general.