The quasi-homogeneity criterion for 3-syzygy plane curves

Let $C=V(f)$ be a reduced $3$-syzygy plane curve, meaning that it has a first syzygy matrix $M_f$ that is a square matrix of size $3$. Let $p\in\operatorname{Sing}(C)$ be an isolated singular point. The 3-syzygy quasi-homogeneity conjecture. The point $p$ is quasi-homogeneous if and only if

$$\operatorname{rk} M_f(p)\geq 1.$$

For free and nearly free curves, analogous criteria relate quasi-homogeneity of singularities to the rank of a first syzygy matrix; the conjecture proposes that the same characterization extends to reduced 3-syzygy plane curves, based on the examples tested in the paper.