Inverse Kazhdan-Lusztig polynomials of matroids need not be log-concave

For every prime power q >= 5, we construct a simple F_q-representable matroid of rank 9 whose inverse Kazhdan-Lusztig polynomial has positive coefficients a_0(q), a_1(q), a_2(q), a_3(q) and satisfies a_2(q)^2 < a_1(q)a_3(q). This disproves the log-concavity conjecture of Gao and Xie. The construction deletes a projective plane and a five-dimensional affine box from PG(8,q). Every nonzero upper interval in the resulting lattice of flats is a full subspace lattice, allowing an explicit computation by rank-summed incidence-algebra inversion. The example over F_5 has 487226 elements. Moreover, a_2(q)^2/(a_1(q)a_3(q)) is asymptotic to q^(-1), so the failure becomes arbitrarily strong while the rank remains fixed. Preliminary research record, version 1. This record is being made public before journal submission and has not undergone external peer review. Later versions may revise the exposition. The attached PDF contains the complete statements, proofs, exact computations, and references for this version.