Strong Donagi–Morrison conjecture for Lazarsfeld–Mukai bundles

Let $(S,H)$ be a polarized K3 surface of genus $g$, let $C\in|H|$ be a smooth irreducible curve, and let $A$ be a complete basepoint-free $g^r_d$ on $C$ with $r\geq 2$ and $\rho(g,r,d)<0$. Let $E_{C,A}$ denote the associated Lazarsfeld–Mukai bundle.

Strong Donagi–Morrison conjecture. There is a nontrivial line bundle $N\hookrightarrow E_{C,A}$ with $h^0(S,N)\geq 2$ such that $E_{C,A}/N$ is stable.

This stronger formulation would produce the line bundles used in known lifting arguments by requiring a stable quotient of the Lazarsfeld–Mukai bundle. The source states that the conjecture is false, with counterexamples cited in the literature.