Universal non-Poissonian pair correlation conjecture for sequences of maximal additive energy

Let $\left(a_n\right)_{n\geq 1}$ be a sequence of distinct integers, and let $A_N$ denote its first $N$ elements. The additive energy of a finite set $A$ of reals is

$$E\left(A\right)=\sum_{a+b=c+d}1,$$

where the sum runs over all quadruples in $A^4$. The universal non-Poissonian pair correlation conjecture. If

$$E\left(A_N\right)=\Omega\left(N^3\right),$$

then for every $\alpha$ the pair correlation of $\left(\left\{a_n\alpha\right\}\right)_{n\geq 1}$ is not Poissonian.

Bourgain proved non-Poissonian pair correlation for a positive-measure set of $\alpha$, and later work showed that the exceptional set has full measure. The conjecture strengthens this to every $\alpha$.