Additive-combinatorial formulation of the exactly-$N$ conjecture

Let $\rho^{\angle}_3(N)$ denote the maximum density of a three-variable corner-free subset in the additive-combinatorial setting used by the paper. Corner-free density conjecture. One should have

$$\rho^{\angle}_3(N) \ge 2^{-o(\sqrt{\log N})}.$$

Possibly even

$$\rho^{\angle}_3(N) \ge 2^{-(\log\log N)^{O(1)}}.$$

These bounds are presented as the additive-combinatorial translation of the conjectured improvement in the NOF communication complexity of exactly-$N$; the precise definition and ambient domain of $\rho^{\angle}_3(N)$ should be checked in the paper.