In 1987 Harris proved-among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular partial sums S42A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums S4nAMA f ,nA ∈{1, 2, ...,mA − 1} on unbounded Vilenkin groups converge almost everywhere to f for each function f ∈ L2.