The supersolvable plane curve freeness conjecture

Let $C$ be a reduced plane curve in $cmathbb{P}^2$. A point $p\in C$ is modular if the central projection from $p$ induces a locally trivial fibration of the complement $\mathbb{P}^2\setminus C$ over $\mathbb{P}^1$, and $C$ is supersolvable if it has at least one modular point. Supersolvable curve conjecture. Every supersolvable plane curve is free. The assertion generalizes the corresponding fact for supersolvable line arrangements; it was stated as an open conjecture and the source gives no general resolution.