Sharpness and convergence conjecture for two-dimensional Vilenkin-Fourier sums

Let $H_p^{\square}(G_m^2)$ denote the two-dimensional martingale Hardy space and let $M_k$ be the associated scale sequence. For $f$ in these spaces, write $\omega_{H_p^{\square}(G_m^2)}(1/M_k,f)$ for the corresponding modulus of continuity. The operator $S_{m,n}f$ denotes the $(m,n)$-th partial sum of the two-dimensional Vilenkin-Fourier series.

a) Convergence conjecture. If $f\in H_p^{\square}(G_m^2)$, $2^{-\alpha}\leq m/n\leq 2^{\alpha}$, and

$$\omega_{H_{1}^{\square}(G_m^{2})}\left(\frac{1}{M_k},f\right)=o\left(\frac{1}{k^2}\right)\quad\text{as }k\to\infty,$$

then

$$\left\Vert S_{m,n}f-f\right\Vert_{H_{1}^{\square}(G_m^{2})}\to0\quad\text{as }m,n\to\infty.$$

b) Sharpness conjecture. If $2^{-\alpha}<m/n\leq2^{\alpha}$, then there exists a martingale $f\in H_{1}^{\square}(G_m^2)$ such that

$$\omega_{H_{p}^{\square}(G_m^{2})}\left(\frac{1}{M_k},f\right)=O\left(\frac{1}{k^2}\right)\quad\text{as }k\to\infty$$

and

$$\left\Vert S_{m,n}f-f\right\Vert_{1}\nrightarrow0\quad\text{as }m,n\to\infty.$$

These two assertions are presented together as an open problem: the first gives a sufficient modulus-of-continuity condition for convergence, while the second asserts that the $1/k^2$ threshold is sharp. The source supplies no resolution.