Stokolos's reformulation of Zygmund's conjecture for translation-invariant rectangle bases

Let $\mathcal{I}^n$ denote the collection of axis-parallel rectangles in $\mathbb{R}^n$, and let $\mathcal{B}\subseteq\mathcal{I}^n$ be a family invariant under translations. The maximal operator associated with $\mathcal{B}$ is denoted by $M_{\mathcal{B}}$. Say that it is sharply bounded from $L^1(1+\log^+L^1)^r$ to $L^{1,\infty}$ when it satisfies the corresponding weak-type estimate and no smaller Orlicz growth suffices.

Stokolos's reformulation of Zygmund's conjecture. There exists an integer $k$ with $1\leq k\leq n$ such that $M_{\mathcal{B}}$ is sharply bounded from $L^1\left(1+\log^+L^1\right)^{k-1}$ to $L^{1,\infty}$.

This reformulation was proposed after the parametrized version was disproved and is motivated by Stokolos's theorem in dimension two. The supplied text gives no resolution of the reformulated statement, so it remains open here.