Let $P(n)$ be the statement that 
$$n < \mathrm{rad}(n(n-1)(n-2)),$$
where $\mathrm{rad}$ is the [radical of an integer,](https://en.wikipedia.org/wiki/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?