Let $P(n)$ be the statement that $$n < \mathrm{rad}(n(n-1)(n-2)),$$ where $\mathrm{rad}$ is the radical of an integer, that is defined as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p$$ for integers $m>1$ with $\operatorname{rad}(1)=1$.

I checked that $P(n)$ holds for $3 \le n \le 3.10^7$.

My question: Is $P(n)$ true for any positive integer $n \geq 3$?

Also, is this a pre-existing conjecture?