Microlocal multiplicity conjecture for matroids

Let $M$ be a loopless matroid of rank $d$. Its microlocal multiplicity $m_M$ is defined by

$$m_M=(-1)^{d} \sum_{F\in \mathcal{L}(M)}2^{\operatorname{rk} F}\chi_{M^F}(1/2)P_{M_F}(1).$$

Here $\mathcal{L}(M)$ is the lattice of flats of $M$, $\chi_M$ is its characteristic polynomial, $P_M$ is its Kazhdan–Lusztig polynomial, and $M^F$ and $M_F$ are respectively the localization and contraction of $M$ at $F$.

Microlocal multiplicity conjecture. For every loopless matroid $M$,

$$m_M\geq 0.$$

Moreover, $m_M=0$ if and only if $M$ has a Boolean summand, meaning that $M$ is a direct sum of a Boolean matroid and another arbitrary matroid.

For realizable matroids, nonnegativity follows from the effectiveness of the characteristic cycle of the intersection complex of the associated matroid Schubert variety. The conjecture extends this positivity statement to arbitrary loopless matroids and predicts precisely when the multiplicity vanishes.