Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: {\it For a smooth projective curve $C$ of genus $g$ in characteristic 0, the condition ${\rm Cliff} C>l$ is equivalent to the fact that $K_{g-l'-2,1}(C,K_C)=0, \forall l'\leq l$.} We propose a new approach, which allows up to prove this result for generic curves $C$ of genus $g(C)$ and gonality ${\rm gon(C)}$ in the range $$\frac{g(C)}{3}+1\leq {\rm gon(C)}\leq\frac{g(C)}{2}+1.$$