Strong summability conjecture for two-dimensional Vilenkin-Fourier series

Let $H_1(G_m^2)$ be the two-dimensional martingale Hardy space, and let $S_{k,l}f$ denote the $(k,l)$-th partial sum of the two-dimensional Vilenkin-Fourier series. Fix $\alpha$ and sum over indices satisfying $2^{-\alpha}\leq k/l\leq2^{\alpha}$ and $(k,l)\leq(n,m)$ coordinatewise.

Strong summability conjecture. There exists an absolute constant $c$ such that

$$\sup_{n,m\geq2}\frac{1}{\log n\log m}\sum_{\substack{2^{-\alpha}\leq k/l\leq2^{\alpha},\\(k,l)\leq(n,m)}}\frac{\left\Vert S_{k,l}f\right\Vert_{H_1(G_m^2)}}{kl}\leq c\left\Vert f\right\Vert_{H_1(G_m^2)}.$$

The paper explicitly describes this strong summability estimate in the $p=1$ case as an open problem. It would give a uniform logarithmically normalized bound for partial sums in a restricted cone of indices.