We present a critical examination of the ideal MHD model far the stationary Crab pulsar wind, which has a terminal flow Lorentz factor gamma(infinity) similar to 10(6) and may have a terminal ratio of Poynting flux to kinetic energy flux as low as sigma similar to 10(-2)-10(-3). We first show that transitions to a low-sigma configuration cannot occur gradually in regions well beyond the light cylinder where the flow has already become ultrarelativistic. This is because the poloidal field lines do not expand sufficiently beyond the fast critical point to convert the electromagnetic energy into flow kinetic energy. As an alternative, we consider whether the acceleration may proceed abruptly, with the flow rapidly passing through the fast critical point and expanding into a low-sigma configuration. Such rapid expansion of held lines, analogous to that in a de Laval nozzle, requires the poloidal fields to be highly compressed upstream of the fast critical point. We categorize the rapid field expansion generally into two prototypes: sheetlike and fountain-like, with the former being a spontaneous transition and the latter requiring external pressure supports. Unfortunately, it is shown that both types of rapid acceleration fail to satisfy either the energy and momentum conservation or the MHD flux-freezing condition. Nevertheless, we have pinned down the only situation where a stationary, ideal-MHD low-sigma wind may exist. It requires almost the entire wind acceleration to occur in the immediate neighborhood of the light cylinder. Moreover, it also demands drastic modifications to the conventional picture of the pulsar dipole magnetosphere, in that the outer magnetosphere must be dominated by the toroidal fields and that the pulsar wind is carried by only a small fraction of the magnetospheric field lines emerging from the star.