The square-root interval conjecture for the prime number theorem

Let $p_k$ denote the $k$th prime, let $s_k:=\{p_k^2,\dots,p_{k+1}^2-1\}$, and let $\pi_k$ be the number of primes in $s_k$. For $x$ and an interval length $y=y(x)$, write $\pi(x+y)-\pi(x)$ for the number of primes in $[x,x+y]$. Square-root interval conjecture. The choice $y=x^{1/2}$ is necessary and sufficient for

$$\pi(x+y)-\pi(x)\sim\frac{y}{\log(x+y)}$$

to hold for all $x$ as $x\to\infty$. This proposes $x^{1/2}$ as the sharp threshold for a prime number theorem valid in every interval, contrary to earlier conjectures placing the threshold near $x^\epsilon$; the paper presents heuristic and computational evidence, but no proof.