The rational nearly cuspidal curve bound for the Jacobian module

Let $C:f=0$ be a plane curve, and let $J_f e S$ be the Jacobian ideal generated by the \partial derivatives of $f$. Let $I_f$ be the saturation of $J_f$ with respect to the maximal ideal ${\bf m}=(x,y,z)$, and define the Jacobian module

$$N(f)=I_f/J_f.$$

Set $n(f)_k=\dim N(f)_k$ and

$$\nu(C)=\max_j\{n(f)_j\}.$$

Suppose that $C$ is rational and nearly cuspidal. Rational nearly cuspidal curve conjecture. One has

$$\nu(C)\leq 2.$$

The bound is motivated by the study of curves with $\nu(C)=2$, for which the source proves structural results depending on the parity of the degree. The supplied text does not state whether this conjectural bound has been proved or disproved in full.