The generalized Bayer–Macrì–Toda inequality for stable objects

Let $X$ be the threefold considered in the paper, let $H$ be the chosen ample divisor, and let $\Gamma\in\operatorname{Hom}_{\mathbb{Q}}(\mathrm{CH}^1_{\mathrm{num}}(X)_{\mathbb{Q}},\mathbb{Q})$ satisfy $\Gamma(H)\geq0$. For $b\in\mathbb{R}$ and $v\in\Lambda$, define

$$v_i^b=\sum_{j=0}^i\frac{(-b)^j}{j!}v_{i-j}.$$

Let $\mathcal{U}$ be the parameter domain, $\mathcal{A}^b(X)$ the tilted heart, and $\nu_{b,w}$ the tilt slope. Generalized BMT conjecture. If $(b,w)\in\mathcal{U}$ and $E\in\mathcal{A}^b(X)$ is $\nu_{b,w}$-stable with $\nu_{b,w}(E)=b$, then

$$\mathbf{v}_3^b(E)\leq\frac{2w-b^2}{6}\mathbf{v}_1^b(E)+\Gamma.\mathbf{ch}^{\gamma}_1(E)-b\frac{\Gamma.H}{H^3}\mathbf{v}_0(E).$$

This is the singular-scheme version of the inequality first proposed in the smooth case; the source provides no resolution of the conjecture.