This paper considers the efficiency of a contestable natural monopoly if consumers are heterogeneous and the monopolist can differentiate prices imperfectly. With restricted entry, the standard result in this case is that the monopoly offers a menu of price-quantity combinations which leads to the well-known 'no-distortion-at-the-top` pricing. Low demand consumers are induced to consume less than their first-best quantity, while high demand consumers buy a quantity where their marginal willingness to pay equals marginal cost. The paper shows that this type of inefficiency may also appear in a contestable market. Depending on cost and demand structures, first best efficiency can al so be a sustainable equilibrium. However, due to the existence of a continuum of equilibria, first best efficiency is never guaranteed. Most notably, even a stable 'distortion-at-the-top' result is possible.