  In a recent work, Kaiser, Stehl\'ik and \v{S}krekovski provide a family of
critically 3-chromatic graphs whose expansions do not result in critically
4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco,
Ha and Van Tuyl. The cover ideal of the smallest member of this family also
gives a counterexample to the persistence and non-increasing depth properties.
In this paper, we show that the cover ideals of all members of their family of
graphs indeed fail to have the persistence and non-increasing depth properties.
