  In previous works, the authors investigated the relationships between linear
stability of a generated linear series $|V|$ on a curve $C$, and slope
stabillity of the vector bundle $M_{V,L} := \ker (V \otimes \mathcal{O}_C \to
L)$. In particular, the second named author and L. Stoppino conjecture that,
for a complete linear system $|L|$, linear (semi)stability is equivalent to
slope (semi)stability of $M_V$, and the first and third named authors proved
that this conjecture holds for hyperelliptic and for generic curves.
  In this work we provide a counterexample to this conjecture on any smooth
plane curve of degree $7$.
