The Ciliberto-Di Gennaro conjecture for $d=5$

The Ciliberto-Di Gennaro conjecture predicts that a nodal hypersurface of degree $d\geq 3$ with at most $2(d-2)(d-1)$ nodes is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has $2(d-2)(d-1)$ nodes. This conjecture is classically known for $d=3,4$. In 2022 the author proved this conjecture for $d\geq 7$ by the author. Kvitko announced a proof for $d=6$ in 2025. In this paper we prove the conjecture for the remaining open value of $d$, namely $d=5$.