Characterize the objects parametrized by Chow and pure cohomology groups

Characterize the combinatorial or geometric objects parametrized by the Chow groups and pure cohomology groups of closed Richardson varieties, positroid varieties, and brick varieties.

Background

The paper establishes that the Chow groups and lowest-weight rational Borel–Moore homology groups of closed Richardson varieties, positroid varieties, and brick varieties are generated by classes of closed strata. It asks for a more informative parametrization of these groups, analogous to the parametrization of Schubert-variety Chow groups by elements of Bruhat order. The authors suggest that the positroid case may involve suitable collections of comparable $k$-Bruhat intervals and that brick-variety parametrizations may arise from subwords of the defining braid.