Characterization of nonreducible associative idempotent monotone functions

Let $X$ be a totally ordered Abelian group with respect to addition, and let $F:X^n\to X$ be an associative, idempotent, monotone function. The function $F$ is reducible if there exists an associative binary function $G:X^2\to X$ such that $F$ is obtained by iterating $G$; otherwise it is nonreducible. Nonreducibility conjecture. The function $F$ is not reducible if and only if $n$ is odd and there exists a monotone bijection $g:X\to X$ such that

$$F(x_1,x_2,\dots,x_n)=g^{-1}\left(\sum_{i=1}^n(-1)^i g(x_i)\right).$$

This would characterize the associative idempotent monotone functions that are not reducible, extending the known reducibility results beyond the nondecreasing case. The source presents the characterization as an open conjecture.