Full rank of the max-min set-composition matrix

Prove that, for each fixed d, the matrix whose rows are indexed by loopless nested matroids on [d] and whose columns are indexed by max-min set compositions of [d], with entries recording whether a set composition is generic for the corresponding matroid, has full rank d!.

Background

Max-min set compositions are set compositions $(\pi_1,\ldots,\pi_k)$ satisfying $\max \pi_i>\min \pi_{i+1}$ for every adjacent pair. The paper proves that these set compositions are in bijection with permutations, so there are $d!$ of them.

The proposed matrix tests genericity of each max-min set composition for each loopless nested matroid. Full rank would provide an explicit certificate for the conjectured linear independence of the chromatic word-quasisymmetric functions of loopless nested matroids.