Bayer–Macrì–Stellari quadratic Bogomolov–Gieseker inequality conjecture

Let $X$ be a smooth projective 3-fold. Let $\mathcal{B}_{\omega,B}$ be the tilted heart and let $\nu_{\omega,B}$ be the associated tilt slope. For an object $E\in\mathcal{B}_{\omega,B}$, let $\overline{\Delta}_{\omega,B}(E)$ and $\overline{\nabla}_{\omega,B}(E)$ denote the quadratic expressions defined in the source, with

$$\overline{\nabla}_{\omega,B}(v)=2(v_2^B)^2-3v_1^Bv_3^B.$$

Bayer–Macrì–Stellari quadratic Bogomolov–Gieseker inequality conjecture. For every $\nu_{\omega,B}$-semistable object $E\in\mathcal{B}_{\omega,B}$,

$$\overline{\Delta}_{\omega,B}(E)+6\overline{\nabla}_{\omega,B}(E)\geq 0.$$

This is an equivalent quadratic form of the Bogomolov–Gieseker inequality conjecture and is intended to establish Bridgeland stability conditions on smooth projective 3-folds. The source presents it as conjectural.