Syzygies of 5-gonal Canonical Curves

We show that for $5$-gonal curves of odd genus $g\geq 13$ and even genus $g\geq 28$ the $\lceil \frac{g-1}{2}\rceil$-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree $5$.