I have an {0,1}, invertible, triangular matrix, that I would like to show to be totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?

Seymour's decomposition theorem for regular matroids yields a polynomialtime algorithm for testing if any {0,1,1} matrix is totally unimodular. Unfortunately, due to the sound of paint drying on this question of Gordon Royle, it seems as if no one has yet implemented it. 

