A result of Mozes-Shah (1995) implies that a sequence of distinct totally geodesic surfaces equidistributes in a closed hyperbolic 3-manifold. Is this also true for a sequence of closed, connected, essential K-quasifuchsian surfaces with K going to one? Not necessarily — we will construct a family of increasingly geodesic connected quasifuchsian surfaces that can accumulate on any finite collection of totally geodesic surfaces. To explain that, we will outline the construction of quasifuchsian surface subgroups, following Kahn, Markovic and Wright. We will also use ideas of Liu-Markovic that ensure that our surfaces are always connected.