Sum rules for the special values of fully packed loop polynomials

For a matching $\pi$, define $G_\pi=A_\pi(-|\pi|)$; let $A_n$ and $A_n^V$ denote the enumerative quantities used in the source, and let $^n$ denote the empty matching with $n$ nested arches. Sum-rule conjecture. For every positive integer $n$,

$$\sum_{\pi:|\pi|=n}|G_\pi|=A_n,\qquad \sum_{\pi:|\pi|=n}G_\pi=(-1)^{\frac{n(n-1)}2}(A_n^V)^2,$$

and

$$G_{^n}=\begin{cases} (-1)^{\frac{n(n-1)}2}(A_{n+1}^V)^2,&n\text{ is even},\\ (-1)^{\frac{n(n-1)}2}A_n^VA_{n+2}^V,&n\text{ is odd}.\end{cases}$$

These identities describe striking global relations among the special values $G_\pi$; the source notes that part of the second equality follows from other conjectures, while the general conjectural picture remains open.