Failure of Generic Green's conjecture in characteristic 3

Let $g$ denote the genus of a curve over a field of characteristic $3$. Generic Green failure conjecture. Generic Green's conjecture fails in characteristic $3$ for genera $9$ and $10$.

The claim arises from computations in the paper, where the closest observed diagram to the generic one in characteristic $3$ has a nonzero sequence in these genera. It is presented as a naive computationally motivated assertion rather than as a proved theorem.