The link-pattern balance conjecture for the O(1) loop model

A link-pattern is a noncrossing pairing of the $2n$ vertices, and $A_n(π)$ denotes the number of fully packed loop states having link-pattern $π$. For each $i=1,2n$, let $h_i$ be the operation that reconnects the links incident to vertices $i$ and $i+1$ as described in the paper, and let $Π_i(π)$ be the set of link-patterns $π'$ such that $h_i(π')=π$.

Link-pattern balance conjecture. For any $n=1,2,\ldots$, one has

$$\sum_{i=1}^{2n}\sum_{\pi'\in\Pi_i(\pi)}A_n(\pi')=2nA_n(\pi).$$

Equivalently, in the associated game, a uniformly selected FPL state and a uniformly selected operation $h_i$ give equal probabilities of producing any fixed target link-pattern. The conjecture is illustrated for $n=4$ in the source; its general status is not established there.