Counterexamples to "Transitive Regret"

Theorem 1 in Bikhchandani & Segal (2011; Theoretical Economics) suggests that a complete, transitive, monotonic, and continuous preference is regret based if and only if it is expected utility. Their Proposition 1 suggests that transitivity and continuity of a regret-based preference implies an equivalence condition: if random variables X and Y have the same distribution, then X~ Y. We give counterexamples to Proposition 1.