A remark on weighted Bounded Negativity for blow-ups of the projective plane

Motivated by the weighted Bounded Negativity Conjecture, we prove that all but finitely many reduced and irreducible curves $C$ on the blow-up of ${\mathbb{P}}^2$ at $n$ points satisfy the inequality $C^2\ge \min \{-\frac{1}{12}n(C\mathrm{.}L+27),-2\}$, where $L$ is the pull-back of a line. This partially improves on some result by Laface and Pokora.