Let $n\ge 3$ be an integer. I would like to know if the following property $(P_n)$ holds: for all real numbers $a_i$ such that $\sum\limits_{i=1}^na_i\geq0 $ and $\sum\limits_{1\leq i<j<k\leq n}a_ia_ja_k\geq0$, we have $$n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3.$$ I have a proof that $(P_n)$ holds for $3\leq n\leq8$, but for $n\geq9$ my method does not work and I did not see any counterexample for $n\ge 9$.

Is the inequality $(P_n)$ true for all $n$? Or otherwise, what is the largest value of $n$ for which it holds?

Thank you!