Real-root conjecture for the tau-deformed groundstate polynomials

Let $\pi$ be a matching, and consider the bivariate polynomial $\Psi_\pi(\tau,t)$ as a polynomial in $t$ with coefficients in $\mathbb{Q}[\tau]$. Let $d(\pi)$ and $m_i(\pi)$ have the meanings used for the matching $\pi$. Tau-deformed real-root conjecture. Its real roots in $t$ are the negative integers $-p$, with multiplicity $m_p(\pi)$; equivalently,

$$\Psi_\pi(\tau,t)=\frac{1}{|d(\pi)|!}\prod_{i=1}^{|\pi|}(t+i)^{m_i(\pi)}Q_\pi(\tau,t),$$

where $Q_\pi(\tau,t)$ is a polynomial in $t$ with no real roots. This extension was verified for all $\Psi_\pi(\tau,t)$ with $|\pi|\leq8$, while the general claim remains open.