The local square-energy conjecture in logarithmic intervals

For a finite set $E$ of squares in the short interval

$$E\subset\{k^2:N\le k\le N+N/(\log N)^\delta\},$$

let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The local energy conjecture. There exists $\delta>0$ such that

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

The source introduces this as the special square-polynomial case of the short interval conjectures and states that it remains open.