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

Let $C\subset \PP^2$ be a reduced irreducible complex plane curve with complement $U$. In this paper we prove that $χ(U)\geq ν(C)$, where $ν(C)$ is the freeness defect of $C$ defined in \cite{Dim24}. When $C$ is a rational curve with unibranched singularities, this inequality proves the conjecture