Finite-time wave breaking criterion for the generalized 0-Holm-Staley equation

Let $k=1$, let $u_0\in H^s(\mathbb{R})$ with $s>3/2$, and let $u$ be the corresponding solution of equation (8.0.1), with lifespan $T$. Finite-time wave breaking conjecture. The solution $u$ breaks at finite time if and only if

$$\lim_{t\nearrow T}|u_x(t,x)|=+\infty.$$

The authors present this as a provocation because they prove a necessary condition for wave breaking but are unable to find sufficient conditions. The conjecture therefore remains open in the source.