The Effective Gonality Conjecture for syzygies of curves

Let $C$ be a smooth complex algebraic curve of genus $g$ and gonality $k$, and let $L$ be a very ample line bundle on $C$ with

$$\operatorname{deg}(L)\geq 2g-1+k.$$

Write $h^0(L)=\dim H^0(C,L)$, and let $K_{p,q}(C,L)$ denote the Koszul cohomology group of $p$-th order syzygies of weight $q$. Effective Gonality Conjecture. For every such line bundle $L$,

$$K_{h^0(L)-k,1}(C,L)=0.$$

The conjecture gives an effective form of the Gonality Conjecture, strengthening the known asymptotic vanishing for arbitrary smooth curves. The source establishes it in several cases, but the statement is presented as a conjecture in general.