Conjecture on the support of topological limiting measures

Let ${\bf M}$ be a closed hyperbolic 3-manifold with a totally geodesic surface. Let $h_0=h_0({\bf M})$ be the genus of a totally geodesic surface of smallest genus in ${\bf M}$, and let ${\mathcal H}_{h_0}$ denote the corresponding subset of the space of surfaces. Support conjecture. Every topological limiting measure is supported on ${\mathcal H}_{h_0}$. The preceding theorem says that every topological limiting measure is totally scarring, while this conjecture predicts the more precise genus of the totally geodesic surfaces carrying its support. Its resolution is not supplied here.