The Geometric Syzygy Conjecture in Even Genus

Let $C$ be a general curve of genus $g=2k$, let $\omega_C$ be its canonical bundle, and let $K_{p,q}(C,\omega_C)$ denote its Koszul cohomology groups. A linear syzygy is geometric if its rank is at most two more than its homological degree; thus the geometric syzygies in $K_{k-1,1}(C,\omega_C)$ have rank at most $k+1$.

The Geometric Syzygy Conjecture in Even Genus. The last linear syzygy space

$$K_{k-1,1}(C,\omega_C)$$

is spanned by geometric syzygies.

This conjecture gives a geometric refinement of the vanishing statement in Green's conjecture for even genus. It is presented in the source as a folklore conjecture, and no resolution is supplied there.