So-called Whittle networks have recently been shown to give tight approximations for the performance of non-locally balanced networks with blocking, including practical routing policies such as joining the shortest queue. In the present paper, we turn the attention to networks without blocking. To this end, we consider a set of "insensitive" dynamic load balancing schemes preserving the structure of Whittle networks in the case of infinite buffers and examine their efficiency. Using Hausdorff’s theorem, we prove that the optimal insensitive schemes are static in this case, i.e., routing decisions do not depend on the current state of the queues. On the other hand, simulations show that the performance of static policies is generally much worse than that of "non balanced" sensitive dynamic policies. This demonstrates that strict insensitivity and efficiency may be incompatible objectives for networks with dynamic load balancing in case of infinite buffers.