Green's conjecture for canonical curves

Let $X\subseteq \mathbf{P}^{g-1}$ be a non-hyperelliptic canonical curve of genus $g$. Its Clifford index is

$$\operatorname{Cliff}(X)=\min\{d(\varphi)-2r(\varphi)\mid \varphi:X\to\mathbf{P}^r\text{ is non-constant and }d(\varphi)\leq g-1\}.$$

Here $r(\varphi)$ is the dimension of the linear span of the image and $d(\varphi)$ is the degree of the map. Green's conjecture. The Clifford index of $X$ is equal to the least integer $p$ for which Property $(N_p)$ fails for $X$. This predicts the precise first failure of the linear-syzygy properties of a canonical curve, extending Petri's theorem; its general form remains open, although many cases are known.