Mistretta–Stoppino stability equivalence conjecture for generated linear series

Let $C$ be an irreducible projective smooth complex curve of genus $g\geq 2$, let $L\in\operatorname{Pic}^d(C)$ be a globally generated line bundle, and let $V\subseteq H^0(C,L)$ be a generating subspace. The associated dual span bundle $M_{V,L}$ is defined by

$$0\longrightarrow M_{V,L}\longrightarrow V\otimes\mathcal O_C\longrightarrow L\longrightarrow 0.$$

Mistretta–Stoppino's conjecture. If

$$\deg(L)-2(\dim(V)-1)\leq\operatorname{Cliff}(C),$$

then linear (semi)stability of $(L,V)$ is equivalent to (semi)stability of $M_{V,L}$.

The conjecture is known in several cases, including complete linear series and certain degree and codimension ranges. The paper records no claim that it is resolved in full.