On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves

Let $C\subset {\text{PP}}^2$ be a reduced irreducible complex plane curve with complement $U$. In this paper we prove that $\chi (U)\ge \nu (C)$, where $\nu (C)$ is the freeness defect of $C$ defined in \cite{Dim24}. When $C$ is a rational curve with unibranched singularities, this inequality proves the conjecture of Dimca and Sticlaru, that a rational unibranched plane curve is either free or nearly free.