The logarithmic additive-energy conjecture for subsets of squares

For a set $E\subset\{1^2,\ldots,N^2\}$ of squares, let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The logarithmic energy conjecture. There exists a constant $C>0$ such that

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

The source states this as the corresponding conjecture for arbitrary subsets of the first $N$ squares and leaves it open.