Square-prime short-interval conjecture

Let $\tilde\pi(x)$ denote the counting function for square-primes,

$$\tilde\pi(x) \coloneqq \sum_{m \geqslant 1}\sum_{\substack{m^2p \leqslant x \\ \text{$p$ prime}}}1.$$

Square-prime short-interval conjecture. For all $x \geqslant 2$,

$$\tilde\pi(x+\log^3 x)-\tilde\pi(x)<3\log^2 x.$$

Equivalently, every interval $(x,x+\log^3 x]$ contains fewer than $3\log^2 x$ square-primes. This conjecture is the assumption used by the heuristic algorithm to rule out intervals containing too many odd square-primes; its resolution is not supplied here.