Eventually periodic maps conjecture for stable-region closures

Let an interval translation map be eventually periodic when its critical points eventually become periodic, and let a stable region be a region of stable maps in the parameter space. Eventually periodic maps conjecture. Each eventually periodic map is contained in the closure of some stable region. The conjecture asks whether the maximal possible pattern of critical connections is sufficient for membership in the closure of a stable region; the supplied context states that the available perturbation argument does not prove it.