Green-Lazarsfeld's refinement for general Prym-canonical bundles

Let $C$ be a general curve of genus $g=2d-2\geq6$, let $\eta\in\operatorname{Pic}^0(C)$ be a general line bundle, and let $K_{p,q}(C,L)$ denote Koszul cohomology. Green-Lazarsfeld's refinement. One has

$$K_{d-4,2}(C,K_C\otimes\eta)=0.$$

This is presented as a refinement of the Green-Lazarsfeld Gonality Conjecture and is expected to be sharp; the source gives no resolution status.