The freeness conjecture for rational cuspidal plane curves

Let $\mathcal C$ be a rational cuspidal curve in the plane. Here, rational means that $\mathcal C$ has genus zero, and cuspidal means that all its singularities are cusps. Freeness conjecture. The curve $\mathcal C$ is either free or nearly free. This conjecture proposes a sharp structural dichotomy for rational cuspidal plane curves; the supplied text presents it as an intriguing fact, without stating that it has been proved or disproved.