  In arXiv:0908.1031, Varvaruca and Weiss eliminate the existence of cusps for
a free-boundary problem for two-dimensional water waves under assumptions that
hold for solutions such that $\{u>0\}$ is a "strip-like" domain in the sense of
arXiv:0708.4371. In this paper it is proven that cusps do not exists in the
natural setting for these free-boundary problems. In particular, non strip-like
domains are also allowed. This builds upon recent work on non-existence of
cusps in arXiv:2202.00616.
