Dyadic one-function form of Zygmund's conjecture

Let $\Phi:\mathbb{Z}^{d-1}\to\mathbb{Z}$ be monotone increasing in each argument, and let $\mathcal{Z}_d$ be the collection of dyadic rectangles in $\mathbb{R}^d$ whose sidelengths are

$$2^{m_1}\times\dots\times2^{m_{d-1}}\times2^{\Phi(m_1,\dots,m_{d-1})}.$$

Write $\psi_j(x):=x\log(e+x)^j$. Dyadic Zygmund conjecture. The maximal function $\mathcal{M}_{\mathcal{Z}_d}$ is weak-type $\psi_{d-2}(L)$.

This is the dyadic version of the remaining single-function case after the general parameter conjecture was disproved. The source says that the corresponding continuous formulation is open and focuses on this dyadic conjecture, without claiming authorship.