Generic Green's conjecture for canonical curves

Let $C$ be a generic curve of genus $g$ in the moduli space, canonically embedded in ${\bf P}^{g-1}$ when nonhyperelliptic. For a generic curve, the Clifford index is $\lfloor (g-1)/2\rfloor$, and property $N_p$ is defined by requiring projective normality at $N_0$ and successively linear syzygies through the $p$th step.

Generic Green's conjecture. The generic curve $C$ of genus $g$ satisfies

$$N_{\lfloor (g-3)/2\rfloor}.$$

This is a generic prediction about the syzygies of canonical curves and is presented in the source as a more modest form of Green's conjecture. The source does not provide a resolution status for this claim.