The short square-set energy conjecture

Let $E$ be a set of squares contained in the interval of square roots

$$E\subset\{k^2:N\le k\le N+N^{1-\varepsilon}\}.$$

The short square-set energy conjecture. One should have

$$\sum_m r_{E+E}^2(m)\ll |E|^2.$$

The statement appears among the source's open conjectures for square frequencies; the supplied excerpt does not provide a resolution.