Kernel characterization and dimension of chromatic word-quasisymmetric functions

Prove that the kernel of the map from matroids to chromatic word-quasisymmetric functions is spanned by the valuative relations and the loop-coloop relations, equivalently that the values on loopless nested matroids are linearly independent and that the homogeneous degree-d image has dimension d! for every d.

Background

The paper defines the chromatic word-quasisymmetric function map $\ncPsi$ from matroids to word-quasisymmetric functions and proves that it is valuative. Consequently, valuative relations lie in its kernel. The authors also establish loop-coloop relations in the kernel and show that the image in degree $d$ is generated by the values on loopless nested matroids, yielding an upper bound of $d!$ for the dimension.

The authors prove a lower bound of $2^d-d$ using Schubert matroids and conjecture that the upper bound is attained. Establishing the conjecture would identify all kernel relations and produce a Hopf algebra whose homogeneous pieces have dimensions indexed by permutations.