Firoozbakht's conjecture states that: $p_{k}^{1/k}$ is a strictly decreasing function of $k≥1$. Here $p_{k}$ is the sequence of primes.
I know that is statement is not yet proved. But I am asking on a weaker version:
Show that there exist infinitely many indices $k$ such that $$p_{k+1}^{1/k+1}<p_{k}^{1/k}$$
This question is relevant because I want to see the behavior of this sequence over an infinite set.