Quillen-McCord theorem for persistence finite posets

In this paper, we establish a persistence version of the Quillen-McCord theorem for persistence finite posets. Given a map $f\text{\hspace{0.17em}⁣}:P\to Q$ between persistence finite posets $P$ and $Q$ with weakly $\epsilon$-contractible homotopy fibers, we provide an upper bound for the homotopy commutative interleaving distance between $P$ and $Q$.