Convexity of Whitham's highest cusped wave

We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly $C^{1\mathrm{/}2}$ regularity. The convexity of Whitham’s highest cusped wave had been conjectured by Ehrnstr"om and Wahl'en.