The authors give two counterexamples to a conjecture of J. Verschelde, K. Gatermann and R. Cools [Discrete Comput. Geom. 16, No. 1, 69-112 (1996; Zbl 0854.68111)] and of P. Pedersen. It concerns a connectivity property of the mixed cells of subdivisions for Minkowski sums of polytopes and is of interest in connection with certain algorithms. The first counterexample is in dimension two, the second is in dimension three and refutes the conjecture even for subdivisions induced by liftings.