Mistretta–Stoppino gonality-bound conjecture for non-complete linear series

Let $C$ be an irreducible projective smooth complex curve, let $L$ be a line bundle on $C$, and let $V\subseteq H^0(C,L)$ define a base point free non-complete linear series. Let $M_{V,L}$ be its dual span bundle, defined by

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

Let $\gamma$ denote the gonality of $C$. Mistretta–Stoppino's gonality-bound conjecture. If

$$\deg(L)\leq\gamma\bigl(\dim(V)-1\bigr),$$

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

This conjecture is formulated after counterexamples above the gonality bound. The source gives no resolution status.