The negative-curve characterization of instability for line bundles

Let $X$ be a smooth complex projective surface, let $B,\omega\bigin\operatorname{NS}(X)\otimes\mathbb{R}$ with $\omega$ ample, and let $\sigma_{B,\omega}=(Z_{B,\omega},\mathscr{A}_{B,\omega})$ be the Bridgeland stability condition defined by the tilted heart and central charge above. For a line bundle $L$ on $X$, write $L(C)|_C$ for the restriction to $C$ of $L(C)$. Negative-curve instability conjecture. (a) If $L\in\mathscr{A}_{B,\omega}$, then $L$ is not $\sigma_{B,\omega}$-stable if and only if there exists a curve $C$ of negative self-intersection such that $L(-C)$ is a subobject of $L$ in $\mathscr{A}_{B,\omega}$ and

$$0<\operatorname{Arg} Z_{B,\omega}(L)\leq\operatorname{Arg} Z_{B,\omega}(L(-C))\leq\pi.$$

(b) If $L[1]\in\mathscr{A}_{B,\omega}$, then $L$ is not $\sigma_{B,\omega}$-stable if and only if there exists a curve $C$ of negative self-intersection such that $L(C)|_C$ is a subobject of $L[1]$ in $\mathscr{A}_{B,\omega}$ and

$$0<\operatorname{Arg} Z_{B,\omega}(L[1])\leq\operatorname{Arg} Z_{B,\omega}(L(C)|_C)\leq\pi.$$

This conjecture aims to characterize precisely the mechanisms by which a line bundle can fail to be Bridgeland stable: instability should arise exactly from subobjects associated with curves of negative self-intersection. Its status is unresolved in the supplied source.