The termination conjecture for the affine inverse algorithm

Let $f\in\operatorname{\mathcal{PF}}_{m/n}$, extend $f$ periodically to $\mathbb{N}$ by $f(i+tn)=f(i)$, and construct the injective map $U:\mathbb{N}\to\mathbb{N}$ by the insertion procedure in the source: each $\alpha$ is placed in the leftmost available position satisfying conditions (I) and (II), where condition (I) requires $\alpha$ to lie to the right of each $\alpha-tm$ for $1\le t<\alpha/m$, and condition (II) requires

$$f(i)=\#\{\beta\mid \beta\in(\alpha-m,\alpha),\ U^{-1}(\beta)>i\}$$

when $U(i)=\alpha$.

Affine inverse-algorithm termination conjecture. For the resulting $U$, there exists $N$ such that for every $i\ge N$ and every $t\in\mathbb{N}$,

$$U(i+tn)=U(i)+tn,$$

and, in particular, all values $U(N+j)$ for $1\le j\le n$ have been assigned.

The source calls this algorithm conjectural because termination, equivalently eventual $n$-periodicity, had not been proved, although it had been checked on several examples.