Subexponential NOF communication complexity conjecture for exactly-$N$

In the three-player number-on-forehead (NOF) model, let the exactly-$N$ problem be the task of determining whether the sum of the players' hidden inputs equals $N$. Its communication complexity is measured as a function of $N$. Exactly-$N$ communication conjecture. The NOF communication complexity of exactly-$N$ is $o(\sqrt{\log N})$. Possibly it is much smaller, even as small as $(\log\log N)^{O(1)}$. This conjecture would yield improved lower bounds for the corresponding three-variable corner-free-set density in additive combinatorics, and the paper presents the quantitative additive-combinatorial formulation below as a translation of this claim.