Conjecture on breaking the square-root barrier for deterministic prime decisions

Let $N$ be large, and let $[a,b]$ be an interval contained in $[N,2N]$ whose length is at most $N^{1/2+c}$ for some absolute constant $c>0$. Square-root barrier conjecture. There exists an absolute constant $c>0$ such that one can deterministically decide whether $[a,b]$ contains a prime in time

$$O(N^{1/2-c+o(1)}).$$

This conjecture proposes a faster deterministic decision procedure for prime existence in intervals slightly longer than the square-root scale; the paper presents it as an open direction, and no resolution is supplied in the source.