Geometric syzygy conjecture for general canonical curves

Let $C \subset \mathbb{P}^{g-1}$ be a general canonical curve of genus $g$. A scrollar syzygy is a $k$th syzygy of rank $k+2$, and the corresponding schemes of such syzygies are called spaces of scrollar syzygies. Geometric Syzygy Conjecture. All minimal rank syzygies are scrollar, and the spaces of scrollar syzygies are nondegenerate.

This is the precise formulation given for general canonical curves, refining the preceding geometric-syzygy prediction. The supplied source does not state whether it has been proved or disproved.