Almost everywhere convergence of cone-like restricted two-dimensional Fejér means with respect to Vilenkin-like systems

For the two-dimensional Walsh system, Gát and Weisz proved the a.e. convergence of the Fejér means $\sigma_nf$ of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, $\beta^{-1}\leq n_1/n_2\leq\beta$ is ensured with some fixed parameter $\beta\geq1$. The result of Gát and Weisz was generalized by Gát and the author in the way that the indices are inside a cone-like set.
In the present paper, the a.e. convergence is proved for the Fejér means of integrable functions with respect to two-dimensional Vilenkin-like systems provided that the set of indeces is in a cone-like set. That is, the result of Gát and the author is generalized to a general orthonormal system, which contains as special cases the Walsh system, the Vilenkin system, the character system of the group of 2-adic integers, the UDMD system, and the representative product system of CTD (compact totally disconnected) groups.