Existence of water waves evolving from a graph to a splash singularity

Let $z_0(\alpha)$ and $\omega_0(\alpha)$ be initial data for solutions of the water wave equations. At time $0$, the curve $z_0(\alpha)$ may be parameterized as a graph; the interface can then turn over at a finite time $T_1>0$ and produce a splash at a finite time $T_2>T_1$.

Graph-to-splash existence conjecture. There exist initial data $z_0(\alpha),\omega_0(\alpha)$ of solutions of the water wave equations such that at time $0$ the curve $z_0(\alpha)$ can be parameterized as a graph, the interface then turns over at a finite time $T_1>0$, and finally produces a splash at a finite time $T_2>T_1$.

The statement concerns the existence of water-wave interfaces that evolve from an initially graphical configuration through overturning and then develop a splash singularity. The surrounding discussion places it alongside known constructions of splash singularities and results on their absence in internal waves; the supplied text does not establish whether this particular statement has been resolved.