Zygmund's conjecture I on maximal operators for parametrized rectangle bases

Let $n,k\geq 1$. Let $\{\phi_i:i\leq n\}$ be $n$ positive real functions depending on $k$ variables, increasing in each variable separately and assuming arbitrarily small values. Let $B_{\boldsymbol{\phi}}$ be the basis of all axis-parallel rectangles in $\mathbb{R}^n$ whose side lengths are

$$\phi_1(t)\times\dots\times\phi_n(t),\qquad t\in\mathbb{R}^k.$$

Zygmund's conjecture I. The maximal operator $M_{B_{\boldsymbol{\phi}}}$ is bounded from $L^1\left(1+\log^+L^1\right)^{k-1}$ to $L^{1,\infty}$.

The conjecture was disproved in the stated form by counterexamples of Soria in the case $n=3$, $k=2$, and subsequently by Rey for another class of examples; these results show that this formulation is false, although the underlying idea motivated a reformulation by Stokolos.