Conjecture that geometrical limiting measures are Liouville

Let ${\bf M}$ be a closed hyperbolic 3-manifold. A geometrical limiting measure is a probability measure on ${\mathcal {G}}_2({\bf {M}} )$ obtained as a weak limit of geometrical limiting measures associated to sequences $\epsilon_n\to 0$. The Liouville-measure conjecture. Every geometrical limiting measure is Liouville. This is a conjecture about the limiting distribution of geometrically defined surfaces; the preceding theorem only establishes a positive lower bound for its Liouville part, so the assertion that no totally geodesic component remains is open.