Gaetz–Pierson's real-rootedness conjecture for identity-pattern character polynomials

Let $k\geq 0$ and let $\lambda$ be a partition. The polynomial $a_{\mathrm{id}_k}^\lambda(n)$ is the character coefficient polynomial associated with the identity pattern of length $k$. Gaetz–Pierson's conjecture. For any partition $\lambda$, the polynomial $a_{\mathrm{id}_k}^\lambda(n)$ is real-rooted with all roots less than $k$. This conjecture concerns the root structure of the character polynomials arising from identity-pattern occurrences; its status is not specified in the source.