A counterexample to a claim about stochastic simulations

Engen & Lillegard (1997) presented a general method for doing Monte Carlo simulations conditioned on a sufficient statistic.The basic idea was to adjust the parameter values in the corresponding unconditional simulation so that the actual value of the sufficient statistic is obtained, and the claim was that if this adjustment is unique then the modified simulation is from the conditional distribution. Unfortunately the claim is not correct, as shown by a counterexample.