Existence of Donagi–Morrison lifts when L is not ample

Determine whether, for a K3 surface X with a base point free and big but non-ample line bundle L and a smooth curve C in |L|, and for a base point free line bundle A on C, there exists a line bundle N on X that is a Donagi–Morrison lift of A; namely, N is adapted to |L| (both N and L ⊗ N^∨ have at least two independent global sections and the number of global sections of N restricted to any smooth member of |L| is constant across the linear system) and satisfies that |A| is contained in the restriction of |N| to C and the Clifford index of N ⊗ O_C is at most the Clifford index of A.

Background

Prior works have approached the Donagi–Morrison conjecture when L is ample, including results for nets and characterizations when a linear series computes the Clifford index of the curve. The author notes that extending existence of such lifts to the non-ample case remains unresolved in general, motivating the sufficient conditions proved in Theorem 1.2.