Agents have uniform preferences if a weakly decreasing utility function determines each agent's preference ranking over the same order of alternatives. We show that the impossibility in the random assignment problem between strategyproofness, ordinally efficiency, and fairness in the sense of equal division lower bound, prevails even if agents have uniform preferences. Furthermore, it continues to hold even if we weaken the strategyproofness to upper-contour strategyproofness, or the ordinal efficiency to robust ex-post Pareto efficiency.