The quadratic Bayer–Macrì–Toda inequality for semistable objects

Let $X$, $H$, $\Gamma$, $\mathcal{U}$, $\mathcal{A}^b(X)$, $\nu_{b,w}$, $\mathbf{v}_i^b$, $\overline{\Delta}$, and $\mathbf{ch}^{\gamma}_1$ be as in the paper, with $\Gamma(H)\geq0$. Quadratic BMT conjecture. For every $(b,w)\in\mathcal{U}$ and every $\nu_{b,w}$-semistable object $E\in\mathcal{A}^b(X)$,

$$0\leq Q^{\Gamma}_{b,w}(E),$$

where

$$Q^{\Gamma}_{b,w}(E)=(2w-b^2)\left(\overline{\Delta}(E)+3\frac{\Gamma.H}{H^3}(\mathbf{v}_0(E))^2\right)+2\mathbf{v}_2^b(E)\left(2\mathbf{v}_2^b(E)-3\frac{\Gamma.H}{H^3}\mathbf{v}_0(E)\right)-6\mathbf{v}_1^b(E)\left(\mathbf{v}_3^b(E)-\Gamma.\mathbf{ch}^{\gamma}_1(E)+b\frac{\Gamma.H}{H^3}\mathbf{v}_0(E)\right).$$

This is an equivalent semistable formulation of the generalized BMT inequality; the source gives no resolution.