Schreyer's geometric syzygy conjecture for canonical curves

Let $C$ be a general curve of genus $g$. For each relevant homological index $i$, the linear syzygy space $\mathrm{K}_{i,1}(C,\omega_C)$ consists of syzygies of rank at least $i+1$.

Geometric Syzygy Conjecture. All linear syzygy spaces $\mathrm{K}_{i,1}(C,\omega_C)$ are spanned by syzygies of minimal rank $i+1$.

This is Schreyer's geometric form of the minimal syzygy conjecture: the linear syzygies of a general canonical curve should arise from the rational normal scrolls containing it. The source provides no resolution status, so the conjecture is recorded as open.