Hilbert-function criterion for freeness of plane curves

Let $C:f=0$ be a plane curve of degree $d$. Let $ct(f)$ and $st(f)$ denote the coincidence and stability thresholds of its Milnor algebra, and let $T=3(d-2)$. Freeness criterion conjecture. The curve $C$ is free if and only if

$$ct(f)+st(f)=T.$$

This conjecture proposes a numerical characterization of freeness through the Hilbert function of the Milnor algebra; the source gives no resolution.