  We study the problem of rigidity of closures of totally geodesic plane
immersions in geometrically finite manifolds containing rank $1$ cusps. We show
that the key notion of K-thick recurrence of horocycles fails generically in
this setting. This property was introduced in the recent work of McMullen,
Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups
whose limit sets are circle packings, we derive 2 density criteria for
non-closed geodesic plane immersions, and show that closed immersions give rise
to surfaces with finitely generated fundamental groups. We also obtain results
on the existence and isolation of proper closed immersions of elementary
surfaces.
