Green's canonical syzygy conjecture for generic curves of odd genus

We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2. This completes our previous work, where the Green conjecture for generic curves of genus $g$ with fixed gonality $d$ was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus.