Negative-root multiplicity conjecture for O(n) groundstate polynomials

Let $\pi$ be a matching, let $\psi_\pi(t)$ be its associated polynomial, and let $m_p(\pi)$ denote the multiplicity statistic defined in the source. Let $d(\pi)$ denote the source's associated quantity.

Negative-root multiplicity conjecture. All real roots of $\psi_\pi(t)$ are negative integers, and $-p$ occurs with multiplicity $m_p(\pi)$. Equivalently,

$$\psi_\pi(t)=\frac{1}{\lvert d(\pi)\rvert!}\left(\prod_{p=1}^{\lvert\pi\rvert-1}(t+p)^{m_p(\pi)}\right)Q_\pi(t),$$

where $Q_\pi(t)$ is a polynomial with integer coefficients and no real roots.

The conjecture predicts a precise factorization of the groundstate polynomials according to the nesting statistic. The paper proves only a weaker statement in the parameter-dependent setting, namely that $\psi_\pi(\tau,-p)=0$ whenever $m_p(\pi)\ne0$; the full factorization remains open in the supplied text.