Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes $p,q$, there exists some integer $n$ such that $P(n)=p$ and $P(n+1)=q$?

Problem 6 in the 12th Romanian Master of Mathematics Competitions in 2020 was to prove that there exist infinitely many odd primes $p$ such that, for every $n$, $P(n)P(n+1)\neq 2p$.