Convexity conjecture for Whitham's highest cusped wave

Let $\varphi$ be the profile of a highest cusped periodic traveling-wave solution of Whitham's equation, with speed $\mu>0$, satisfying

$$L\varphi-\mu\varphi+\varphi^2=0,$$

where $L$ has Fourier symbol $m(\xi)=(\tanh\xi/\xi)^{1/2}$. Convexity conjecture for Whitham's highest cusped wave. The profile $\varphi$ is everywhere convex and, near the crest at $0$, satisfies

$$\varphi(x)=\frac{\mu}{2}-\sqrt{\frac{\pi}{8}}|x|^{1/2}+o(|x|).$$

This conjecture concerns the global shape and precise crest asymptotics of the highest cusped wave. The source presents it as a conjecture of Ehrnström and Wahlén; no resolution is supplied in the provided text.