Real-root conjecture for fully packed loop polynomials

Let $\pi$ be a matching, let $A_\pi(t)$ be its polynomial, let $d(\pi)$ denote the relevant degree, and let $m_p(\pi)$ denote the multiplicity assigned to the integer $p$. Real-root conjecture. All real roots of $A_\pi(t)$ are negative integers, with $-p$ occurring with multiplicity $m_p(\pi)$; equivalently,

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

where $Q_\pi(t)$ is an integer-coefficient polynomial with no real roots. The conjecture is supported by exact computations for all matchings of size at most $8$, but no general proof or disproof is given here.