Schreyer's conjecture on extremal canonical syzygies

Let $C$ be a curve of genus $g$ and non-maximal gonality $3\leq k \leq \frac{g+1}{2}$. Assume $W^1_k(C)=\{A\}$ is a reduced single point and $A$ is the unique line bundle of degree at most $g-1$ achieving the Clifford index. Schreyer's conjecture. Then

$$b_{g-k,1}(C,\omega_C)=g-k.$$

This strengthens Green's conjecture for curves with a unique minimal pencil. It has been proved under the bpf-linear growth genericity assumption, so the conjecture is solved.