Real-rootedness of excedance polynomials on 312-avoiding Bruhat intervals

Prove that for every 312-avoiding permutation \(\pi\), the excedance-generating polynomial \(\sum_{\sigma\leq_B\pi}t^{\operatorname{exc}(\sigma)}\) over the lower Bruhat interval \([\mathrm{id},\pi]_B\) is real-rooted.

Background

The paper proves real-rootedness for ascent-generating rook-Eulerian polynomials associated with Ferrers boards, which correspond to lower Bruhat intervals of 312-avoiding permutations. It then asks whether another classical Eulerian statistic, excedance number, has an analogous real-rootedness property on these intervals.

Computational experiments motivate the conjecture for 312-avoiding permutations. The authors explicitly note that the assertion cannot be extended to arbitrary permutations, providing a non-312-avoiding permutation whose corresponding excedance polynomial has non-real roots.