This follows from 'plausible assumptions on the distribution of primes' (as does the question with $k^2$ replaced by $k^d$ for any $d$); the challenge is to prove this unconditionally.

Erdős observed that Cramer's conjecture\[\limsup_{k\to \infty} \frac{p_{k+1}-p_k}{(\log k)^2}=1\]implies that for all $\epsilon>0$ and all sufficiently large $n$ there exists some $k$ such that\[p(n+k)>e^{(1-\epsilon)\sqrt{k}}.\]There is now evidence, however, that Cramer's conjecture is false; a more refined heuristic by Granville [Gr95] suggests this $\limsup$ is $2e^{-\gamma}\approx 1.119\cdots$, and so perhaps the $1+\epsilon$ in the second question should be replaced by $2e^{-\gamma}+\epsilon$.

See also [681] and [682].

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