Conjecture on limiting third twisted Chern character for tilt-stable objects

Let $X$ be a smooth projective 3-fold, fix $B\in\operatorname{NS}(X)_{\mathbb{Q}}$ and an ample class $\omega$, and let $E\in D^b\operatorname{Coh}(X)$. Let $\overline{\beta}(E)$ be the specified root of

$$v_0^B(E)\overline{\beta}^{\,2}-2v_1^B(E)\overline{\beta}+2v_2^B(E)=0,$$

namely

$$\overline{\beta}(E)=\frac{2v_2^B(E)}{v_1^B(E)+\sqrt{\overline{\Delta}_{\omega,B}(E)}}.$$

Limiting BG conjecture. Suppose there is an open neighborhood $U\subset\mathbb{R}^2$ containing $(0,\overline{\beta}(E))$ such that, for every $(\alpha,\beta)\in U$ with $\alpha>0$, the object $E\in\mathcal{B}_{\alpha\omega,B+\beta\omega}$ is $\nu_{\alpha\omega,B+\beta\omega}$-stable. Then

$$\operatorname{ch}_3^{B+\overline{\beta}(E)\omega}(E)\leq 0.$$

This is an equivalent-form conjecture designed to reduce the BG inequality to tilt-stable objects near a limiting wall. The source states it as a generalization of an earlier conjecture, and no resolution is supplied.