Firoozbakht’s conjecture and Cramér's conjecture

Question

Firoozbakht’s conjecture asserts that, for every prime number: $$\sqrt[{k+1}]{p_{k+1}}\lt\sqrt[{k}]{p_k} \ \ \forall k\ge 1$$ Cramer's conjecture asserts that: $$p_{n+1}-p_n=O(\log p_n)^2$$ Firoozbakht’s conjecture is believed to be false, as it contradicts the Cramer's one. Why? Thanks in advance.

Answer 1

Copying my answer to Ratio of logarithmic primes;

This is Firoozbakht’s conjecture. According to the link, it has been verified for primes up to $4\times10^{18}$, but is believed to be false, as it contradicts the Cramér–Granville heuristic.

Added: there is also some discussion which may be useful at https://mathoverflow.net/questions/90327/any-progress-on-the-firoozbakht-conjecture