Dimca–Sticlaru conjectures on rational cuspidal plane curves

Let $S:=\mathbb{C}[x,y,z]$ be the graded polynomial ring, let $f\in S_d$ be homogeneous, and let $C\subset\mathbb{P}^2$ be the reduced plane curve defined by $f=0$. Write $J_f=(f_x,f_y,f_z)$ for the Jacobian ideal, $I_f$ for its saturation with respect to $(x,y,z)$, and $N(f)=I_f/J_f$. The curve $C$ is free when $N(f)=0$, and nearly free when $N(f)\ne 0$ and $\dim N(f)_k\leq 1$ for every $k$. A plane curve is cuspidal when all its singularities are cusps.

Dimca–Sticlaru's conjecture. (i) Any rational cuspidal curve $C$ in the plane is either free or nearly free. (ii) An irreducible plane curve $C$ which is either free or nearly free is rational.

Part (i) is known for rational cuspidal curves of even degree, for curves with abelian complement fundamental group, for curves whose degree is a prime power, and for many unicuspidal curves with a unique Puiseux pair. Part (ii) is contradicted by reducible examples but remains a conjecture for irreducible curves.