On the almost everywhere convergence of sequences of multiple rectangular Fourier sums

In the case when a sequence of $d$-dimensional vectors $\mathrm n_k=(n_k^1,n_k^2,\dots,n_k^d)$ with nonnegative integer coordinates satisfies the condition
$$ n_k^j=\alpha_j m_k+O(1),\quad k\in\mathbb N,\quad1\le j\le d, $$
where $\alpha_1\dots\alpha_d>0$, а $m_k\in\mathbb N$, $\lim_{k\to\infty}m_k=\infty$, under some conditions on the function $\varphi\colon[0,+\infty)\to[0,+\infty)$, it is proved that, if the trigonometric Fourier series of any function from $\varphi(L)([-\pi,\pi))$ converges almost everywhere, then, for any $d\in\mathbb N$ and all $f\in\varphi(L)(\ln^+L)^{d-1}([-\pi,\pi)d)$, the sequence $S_{\mathrm{n}_k}(f,\mathrm x)$ of the rectangular partial sums of the multiple trigonometric Fourier series of the function $f$, as well as the corresponding sequences of partial sums of all of its conjugate series, converges almost everywhere.