Prime-gap bound below exponent 2/7

Determine whether there exists a constant \epsilon<2/7 such that consecutive prime gaps satisfy p_{k+1}-p_k=O(p_{k+1}^{\epsilon}), which would improve the previously known asymptotic upper bound for \Lambda_s without requiring Cramer’s conjecture.

Background

The authors explain that their bound \Lambda_s\leq 8s+O(s^{4/5}d(s)) would improve an earlier O(s^{6/7}) error term if the consecutive-prime gap were O(p_{k+1}^{\epsilon}) for some \epsilon<2/7).

They explicitly state that it is unknown whether this weaker prime-gap estimate holds. It is weaker than Cramer’s conjecture, but is not implied by the prime-gap exponent currently used in the paper.