The polylogarithmic prime-gap conjecture

Let $p_n$ be the $n$th prime, and write $\Delta p_n=p_{n+1}-p_n$ for the gap between consecutive primes.

Polylogarithmic prime-gap conjecture. There exists an absolute constant $k>1$ such that

$$\Delta p_n=O\left((\log p_n)^k\right).$$

The conjecture is explicitly presented as weaker than Cramér's conjecture and would yield explicit sequences of primes growing more slowly than the classical sequences of Mills and Wright. Its status is open.