A modified Donagi–Morrison conjecture for linear series on K3 surfaces

Let $S$ be a $K3$ surface, let $C$ be a smooth irreducible curve of genus $g\geq 2$ on $S$, and let $A$ be a complete, base point free linear series $g^r_d$ on $C$, with $d\leq g-1$ and Brill–Noether number $\rho(g,r,d)<0$. A line bundle $M$ on $S$ is adapted to $|L|$ in the sense used in the paper; the restriction of $M$ to $C$ is $M\otimes\mathcal O_C$. Modified Donagi–Morrison conjecture. There exists a line bundle $M\in\operatorname{Pic}(S)$, adapted to $|L|$, such that $|A|$ is contained in the restriction of $|M|$ to $C$ and

$$\operatorname{Cliff}(M\otimes\mathcal O_C)\leq \operatorname{Cliff}(A).$$

The modification drops the degree inequality that the paper disproves; the candidate is presented as something one might still believe, and no resolution is supplied in the given text.