GIT semistability conjecture for syzygy points of generic canonical curves

Let $C\subset \mathbb{P}^{g-1}$ be a generic canonical curve. For integers $p$ with $p\geq \lfloor (g-3)/2\rfloor$, consider its $(p,2)$-syzygy point $\operatorname{Syz}_{(p,2)}(C)$. Canonical-curve syzygy semistability conjecture. The point $\operatorname{Syz}_{(p,2)}(C)$ is GIT semistable for all $p\geq \lfloor (g-3)/2\rfloor$. Aside from some low-genus cases, this is known for $p=0$ and for $p=1$ for generic curves of odd genus, but remains open in the other stated cases.