Conjecture on linear stability and stability of Lazarsfeld–Mukai bundles

Let $C$ be a curve, let $L$ be a globally generated line bundle on $C$, and let $M_L$ be the kernel bundle in the evaluation sequence

$$0\longrightarrow M_L\longrightarrow H^0(C,L)\otimes\mathcal{O}_C\longrightarrow L\longrightarrow 0.$$

The pair $(H^0(C,L),L)$ is linearly (semi)stable when every generated subseries satisfies the corresponding slope inequality, with strict inequality in the stable case. Linear stability conjecture. The linear (semi)stability of $(H^0(C,L),L)$ is equivalent to (semi)stability of $M_L$. This conjecture seeks to identify linear stability of a complete linear series with slope stability of its Lazarsfeld–Mukai kernel bundle; the source presents it as a conjecture and gives no resolution.