In this talk I will discuss a conjecture of Ehrnström and Wahlén on the profile of travelling wave solutions of extreme form to Whitham's non-local dispersive equation. We will see that there exists a highest, cusped and periodic solution that is convex between consecutive crests, at which $C^{1/2}$-regularity has been shown to be optimal. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.