Complete base-point-free linear-series stability conjecture

Let $C$ be a smooth projective curve and let $L$ be a line bundle such that $V=H^0(C,L)$ gives a complete base point free linear series. Write $|L|$ for this complete linear series and $M_L:=\ker(H^0(C,L)\otimes\mathcal{O}_C\to L)$. Complete-series stability conjecture. Linear (semi)stability of $|L|$ is equivalent to slope (semi)stability of $M_L$. The conjecture has been proved when $C$ is hyperelliptic and when $C$ is Brill–Noether–Petri general; the supplied source therefore records those cases as resolved, while the unrestricted statement is not established there.