Fulman–Kim–Lee–Petersen real-rootedness conjecture for even and odd excedance enumerators

For each positive integer $n$, let $P_{n,k}$ and $Q_{n,k}$ denote the numbers of even and odd permutations in $\mathfrak{S}_n$ with $k$ excedances, respectively. Consider the polynomials

$$\sum_{i=0}^n P_{n,k}t^k \quad\text{and}\quad \sum_{i=0}^n Q_{n,k}t^k.$$

Fulman–Kim–Lee–Petersen conjecture. The polynomials $\sum_{i=0}^n P_{n,k}t^k$ and $\sum_{i=0}^n Q_{n,k}t^k$ are all real-rooted for $n\ge 5$. This is an open problem proposed as part of the study of real-rootedness for excedance enumerators in the alternating subgroup; the supplied text does not provide a resolution.