  We take a closer look at a class of chains with complete connections
introduced by Berger, Hoffman and Sidoravicius. Besides giving a sharper
description of the uniqueness and non-uniqueness regimes, we show that if the
pure majority rule used to fix the dependence on the past is replaced with a
function that is Lipschitz at the origin, then uniqueness always holds, even
with arbitrarily slow decaying variation.
