  We prove that splash-like singularities cannot occur for sufficiently regular
patch solutions to the generalized surface quasi-geostrophic equation on the
plane or half-plane with parameter $\alpha\le \frac 14$. This includes
potential touches of more than two patch boundary segments in the same
location, an eventuality that has not been excluded previously and presents
nontrivial complications (in fact, if we do a priori exclude it, then our
results extend to all $\alpha\in(0,1)$). As a corollary, we obtain an improved
global regularity criterion for $H^3$ patch solutions when $\alpha\le\frac 14$,
namely that finite time singularities cannot occur while the $H^3$ norms of
patch boundaries remain bounded.
