The signed small ball conjecture for multiparameter Haar functions

Let $d\ge 2$. Let $\mathcal D^d$ be the dyadic rectangles in $[0,1)^d$, and let $h_R$ be the associated multiparameter Haar function. For each $n\in\mathbb N_0$, consider rectangles $R\in\mathcal D^d$ with $|R|=2^{-n}$. The signed small ball conjecture. For any choice of signs $\varepsilon_R\in\{-1,+1\}$,

$$\left\|\sum_{R\in\mathcal D^d:\,|R|=2^{-n}}\varepsilon_Rh_R\right\|_\infty\gtrsim n^{\frac d2}.$$

This is the signed form of the small ball problem and predicts a gain of roughly $\sqrt n$ from each dimension in the supremum norm. The one-dimensional estimate discussed in the paper would imply this inequality in all dimensions, but that estimate is not established in the source.