Negative-integer factorization conjecture for fully packed loop polynomials

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 $\pi=\alpha\circ\beta$, where $|\alpha|=p$, and define $G_\alpha=A_\alpha(-|\alpha|)$. Negative-integer factorization conjecture.

$$A_\pi(-p)=G_\alpha A_\beta.$$

This conjecture concerns the special values at negative integers and is supported by computations for all matchings of size at most $8$; its general validity remains open.