Universal constant order fluctuations for the cokernels of block triangular matrices

We prove that for a large class of random block lower triangular matrices, the Sylow $p$-subgroups of their cokernels have the same constant order fluctuations as that of the matrix products studied by Nguyen and Van Peski in arXiv:2409.03099. We also show that the theorem of Nguyen and Van Peski remains true under a weaker assumption on the number of factors in the matrix products.