The single-row root-at-minus-one conjecture for permutation pattern character polynomials

Let $\sigma\in S_k$, and let $a_\sigma^\lambda(n)$ denote the polynomial associated with the stable irreducible-character coefficient for the pattern $\sigma$. A partition $\lambda$ has a single row when it is of the form $(r)$.

Single-row root-at-minus-one conjecture. When $\lambda$ has a single row, the polynomials $a_\sigma^\lambda(n)$ have $n=-1$ as a root.

The conjecture is supported by proofs for the cases $a_{\textnormal{id}_k}^{(1)}$ and $a_{\textnormal{id}_k}^{(2)}$, and by numerical verification for all patterns of size at most four. The source notes that related cases do not all satisfy the same property, so the precise single-row scope is essential.