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Do integrable cellular automata have the confinement property?

We analyse a criterion, introduced by Joshi and Lafortune, for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure. We show that while this criterion can be used in order to single out integrable ultradiscrete systems, there do exist cases where the system is nonintegrable and still the criterion is satisfied. Conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied. We investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be. Finally, we comment on the growth properties of ultradiscrete systems.