Syzygy semistability conjecture for balanced canonical ribbons

Let $k$ be an integer and let $R_{2k+1}$ be the balanced canonical ribbon of genus $2k+1$. For integers $p$ satisfying $2\leq p\leq k-1$, consider its $(p,2)$-syzygy point $\operatorname{Syz}_{(p,2)}(R_{2k+1})$. Balanced-ribbon syzygy semistability conjecture. The syzygy point $\operatorname{Syz}_{(p,2)}(R_{2k+1})$ is semistable for all $2\leq p\leq k-1$. This more precise conjecture is motivated by computer computations for small values of $k$; the source does not establish it in general.