The anatomy-of-integers conjecture for products of integers in short intervals

For a fixed integer $s\geqslant 1$ and integers $h,u$ satisfying $2\leqslant h<u$, let

$${\mathcal R}_s(h,u)=\{(n_1,\ldots,n_s)\in (u,u+h]^s:n_1\cdots n_s=\square\},$$

and write $R_s(h,u)=\#{\mathcal R}_s(h,u)$.

Anatomy-of-integers conjecture. For a fixed integer $s\geqslant 1$ and any integers $2\leqslant h<u$, we have

$$R_{2s}(h,u)\leqslant h^{s+o(1)}$$

uniformly over $u$.

This conjectural bound controls the number of $2s$-tuples from a short interval whose product is a square, and is used to obtain higher-moment estimates for sums of Legendre symbols. The supplied text gives no resolution status or further evidence for the conjecture.