Stage extrema of a chain of safe words with stage-dependent $\varepsilon$-orders agree $2$-adically

For each stage $n$ let $\varepsilon^{(n)}:\mathbb{N}\to\{0,1\}$ be arbitrary and let $\prec_n$ be the associated $2$-adic order on $\mathbb{Z}$ (for $x\neq y$ with $k=\nu_2(y-x)$: $x\prec_n y$ iff the $k$-th two's-complement digit of $x$ equals $\varepsilon^{(n)}_k$). Let $P_0\sqsubseteq P_1\sqsubseteq\cdots$ be finite words of integers, each a prefix of the next, together containing every integer, such that each $P_n$ is safe for its own order $\prec_n$ (the completion "$P_n$, then all other integers in $\prec_n$-order" contains no monotone $4$-term AP). Then for every $k$ there is $N$ such that for all $n\ge N$,

In words: even if the completion order is allowed to change at every stage, the stage extrema of an exhaustive chain of safe words agree modulo every power of $2$ eventually (both are congruent to $\sum_{j<k}\varepsilon^{(n)}_j2^j$ modulo $2^k$). This is a necessary condition on any Ho-style construction of a $4$-AP-free enumeration of $\mathbb{Z}$ with stage-dependent $\varepsilon$-orders; it extends the fixed-order statement and does not by itself rule such constructions out.