Is an Orlicz-Poincaré inequality an open ended condition, and what does that mean?

In [16], Keith and Zhong prove that spaces admitting Poincaré inequalities also admit a priori stronger Poincaré inequalities. We use their technique, with slight adjustments, to obtain a similar result in the case of Orlicz-Poincaré inequalities. We give examples in the plane that show all hypotheses are required.