The Lorenz dominance criterion is the centre piece of inequality analysis. Yet, the appeal of this criterion, which requires considering Lorenz curves in their entirety, is undermined by the practical problem that many sample Lorenz curves intersect in the tails. The commonly used inferential methods, based on central limit theorem arguments, do not apply to the tails since these contain too few observations. By contrast, we propose a test procedure for distributions whose tails lie in the domain of attraction of the Fréchet distribution, which fully takes into account the tail behaviour of Lorenz curves. Our experiments and empirical examples demonstrate the good performance of the proposed test: in many cases we are able to infer that despite sample tail crossings the population Lorenz curves do, in fact, exhibit Lorenz dominance.