Extending recent work of Nath, Saikia, and Sarma on $k$-tuple $\ell$-regular partitions

Let $T_{\ell,k}(n)$ denote the number of $\ell$-regular $k$-tuple partitions of $n$. In a recent work, Nath, Saikia, and Sarma derived several families of congruences for $T_{\ell,k}(n)$, with particular emphasis on the cases $T_{2,3}(n)$ and $T_{4,3}(n)$. In the concluding remarks of their paper, they conjectured that $T_{2,3}(n)$ satisfies an infinite set of congruences modulo 6. In this paper, we confirm their conjecture by proving a much more general result using elementary $q$-series techniques. We also present new families of congruences satisfied by $T_{\ell,k}(n)$.