Is it true that for the optimal distribution $\nu$ (not necessarily uniform) that attains minimum discrepancy $\mathsf{disc}(\mathsf{DISJ}_n)$ for the disjointness function $\mathsf{DISJ}_n$ we have for every rectangle $\mathcal R\subseteq\binom{\{1,2,\dots,n-1,n\}}{\sqrt n}\times\binom{\{1,2,\dots,n-1,n\}}{\sqrt n}$ $$\nu(\mathsf{DISJ}_n^{-1}(0)\cap\mathcal R)\geq\alpha\nu(\mathcal R)-e^{-\alpha n^\beta}$$ for some $\alpha>0$ and $\beta\in(0,1)$?
Distribution attaining minimum discrepancy of disjointness function
communication complexity