Geometric proof of the Chow-ring–positroid Hilbert-series identity

Find a geometric proof, using the variety U_{n,k}, of the identity identifying the Hilbert series of the rational Chow ring CH^*_{\mathbb{Q}}(U_{n,k}) with the codimension-generating function of the positroid strata U_f, namely Corollary \ref{cor:chow-positroid-hilbert}.

Background

The paper constructs the open variety U_{n,k} of n-tuples of lines in \mathbb{P}^{k-1} such that deleting any one line still leaves a spanning configuration, and proves that its rational Chow ring is isomorphic to the algebra R_{n,k,n-k}. The variety U_{n,k} is stratified by positroid strata U_f indexed by strictly bounded affine permutations f, with codimension equal to the affine length \ell(f).

Corollary \ref{cor:chow-positroid-hilbert} therefore gives an equality between the Hilbert series of CH^*{\mathbb{Q}}(U{n,k}) and the generating function \sum_f q^{\ell(f)} over the positroid strata. The paper explicitly states that a geometric explanation of this equality is unavailable. In particular, the seemingly natural strategy of proving linear independence of the fundamental classes [\overline{U_f}] does not work because these classes are usually linearly dependent.