Tau-deformed negative-integer factorization conjecture

Let $\pi$ be a matching and let $p$ be an integer with $1\leq p\leq|\pi|-1$ such that $m_p(\pi)=0$. Write $\pi=\alpha\circ\beta$ with $|\alpha|=p$, and define $G_\alpha(\tau)=\Psi_\alpha(\tau,-|\alpha|)$. Tau-deformed factorization conjecture.

$$\Psi_\pi(\tau,-p)=G_\alpha(\tau)\Psi_\beta(\tau).$$

This conjecture extends the corresponding factorization for $A_\pi(t)$ and was verified for all cases with $|\pi|\leq8$; the general statement remains open.