The zero-correction-function conjecture for the modified Bogomolov–Gieseker inequality

Let $X$ be a smooth projective threefold, let $H\in\operatorname{NS}(X)$ be ample, and let $B\in\operatorname{NS}_{\mathbb{R}}(X)$. Let $A:B+\mathbb{R}\langle H\rangle\to\mathbb{R}_{\geq0}$ be continuous, and suppose the modified Bogomolov–Gieseker inequality holds for $X$ with respect to $A$, with minimal correction constant $\xi(A)>0$. Zero-correction-function conjecture. There exists a continuous function $A':B+\mathbb{R}\langle H\rangle\to\mathbb{R}_{\geq0}$ such that the modified Bogomolov–Gieseker inequality holds with respect to $A'$ and

$$\xi(A')=0.$$

The claim asks whether a positive correction can always be removed by restricting the family of stability parameters through a suitable continuous function; it is open in the source.