The small ball inequality conjecture for hyperbolic Haar sums over large rectangles

Let $d\ge3$, let $n$ be a positive integer, and let $\alpha(R)$ be coefficients indexed by dyadic rectangles $R\subset[0,1]^d$. Write $h_R$ for the tensor-product $L^\infty$-normalized Haar function associated with $R$. The conjectured estimate concerns the sum over rectangles with $|R|\ge2^{-n}$:

$$2^{-n}\sum_{|R|=2^{-n}}|\alpha(R)|\lesssim n^{(d-2)/2}\left\|\sum_{|R|\ge2^{-n}}\alpha(R)h_R\right\|_{L^\infty}.$$

Small ball inequality conjecture. For every $d\ge3$, the displayed inequality should hold. It is a central conjecture in the paper, with consequences for irregularities of distribution, metric entropy, and Brownian-sheet small deviations. The paper establishes bounds with a weaker power of $n$ rather than the conjectured $(d-2)/2$.