Generic geometric syzygy conjecture for canonical curves

Let $C \subset \mathbb{P}^{g-1}$ be a general canonical curve of genus $g$. A $p$-th linear syzygy of $C$ is scrollar if it has rank $p+2$.

Generic geometric syzygy conjecture. For every $p$, the space of $p$-th linear syzygies of $C$ is spanned by scrollar syzygies.

This conjecture proposes that scrollar syzygies account for all linear syzygies of a general canonical curve. The preceding discussion relates scrollar syzygies to linear systems on canonical curves and notes that, for general $k$-gonal canonical curves, Green's conjecture is equivalent to the existence of a scrollar syzygy in every step of the linear strand.