Improving the lower bound N_n ≥ C_{n−1}

Improve the general lower bound N_n ≥ C_{n−1} for the number of ~-equivalence classes of n×n strictly lower triangular matrices over any field of characteristic ≠ 2, seeking a sharper bound valid for all n.

Background

Using the newly introduced invariants (Wall and Measure sequence), the authors prove a lower bound N_n ≥ C_{n−1}. They further show that this bound is strict for n ≥ 4, i.e., N_n > C_{n−1}, which indicates room for strengthening the bound.

They explicitly state that improving this lower bound is an open problem.