Reciprocal identity for Waldschmidt constants of dual matroids

Determine for which matroids M the identity 1/ẑα(I_{Δ(M)}) + 1/ẑα(I_{Δ(M^*)}) = 1 holds, where ẑα denotes the Waldschmidt constant of the Stanley–Reisner ideal of the matroid’s independence complex and M^* is the dual matroid.

Background

The paper shows this reciprocal identity holds for affine geometries (matroids associated to first-order affine Reed–Muller codes) and for sparse paving matroids, based on explicit computations of Waldschmidt constants for I_{Δ(M)} and I_{Δ(M^*)}.

It remains to characterize all matroids for which this reciprocal relation between the Waldschmidt constants of a matroid and its dual is valid.