Higher-rank Mistretta–Stoppino conjecture for Clifford-index bundles

Let $C$ be a smooth projective curve, let $n\geq 2$, and let $E$ be a globally generated vector bundle of rank $n$ computing $\operatorname{Cliff}_n(C)$. Let $M_E$ denote the kernel bundle of the evaluation morphism

$$H^0(E)\otimes\mathcal{O}_C\longrightarrow E.$$

Higher-rank Mistretta–Stoppino conjecture. The following assertions hold: (i) $E$ is linearly (semi)stable; and (ii) $E$ is linearly (semi)stable if and only if $M_E$ is (semi)stable. This extends the rank-one conjecture; the paper verifies the assertions in certain rank-two cases, while the general higher-rank statement remains open.