Multiplicity formula for generalized matroid determinants at k = -1

Determine the multiplicity m_{F,-1} of the factor corresponding to each connected flat F in the factorization of E_{L,-1}=i^*r^*\operatorname{Chow}(L^{-1}), and prove that m_{F,-1}=\mu^+(M/F)\beta(M|_F), where M is the matroid of the linear space L.

Background

For k=-1, the polynomial E_{L,-1}=i^r^\operatorname{Chow}(L^{-1}) is shown to factor into powers of linear forms indexed by connected flats of the matroid M associated with L. The paper establishes its degree and the geometric description of its zero locus, but does not determine the exponents m_{F,-1}.

The authors state that the proposed formula has been verified for uniform matroids and computationally for small non-uniform matroids. The quantities \mu^+(M/F) and \beta(M|_F) are, respectively, the unsigned Möbius invariant of the contraction M/F and the beta invariant of the restriction M|_F.