Bóna's log-concavity conjecture for parking-function polynomials

Let $\mathsf{PF}_n$ be the set of parking functions of length $n$, and define the sum statistic by $\mathsf{\sum}(\pi)=a_1+a_2+\ldots+a_n$ for $\pi=(a_1,a_2,\ldots,a_n)\in\mathsf{PF}_n$. Define the parking-function polynomial

$$P_n(x)=\sum_{\pi\in\mathsf{PF}_n}x^{\mathsf{\sum}(\pi)}.$$

A sequence of real numbers is log-concave if each interior term squared is at least the product of its two neighboring terms; a polynomial is log-concave when its coefficient sequence is log-concave. Bóna's conjecture. The polynomial $P_n(x)$ is log-concave for all $n\geq 1$. This conjecture is recorded as resolved in the supplied status information; log-concavity is stronger than unimodality and belongs to the broader study of log-concavity phenomena for combinatorial and matroid-related invariants.