Complete-series conjecture for linear stability and dual span bundles

Let $C$ be a smooth curve and let ${\cal L}$ be a line bundle on $C$. Let $M_{\cal L}$ denote the kernel bundle of the evaluation map $H^0(C,{\cal L})\otimes\mathcal O_C\to{\cal L}$. Complete-series conjecture. For any curve $C$ and any line bundle ${\cal L}$ on $C$, linear (semi)stability of the complete linear series $(C,{\cal L})$ is equivalent to (semi)stability of $M_{\cal L}$. This is the complete-case counterpart to the conjectures for non-complete linear series; the supplied text gives no resolution, so the conjecture remains open.