Finiteness conjecture for touching points of self-touching CMC surfaces

Let $M$ be a three-dimensional Riemannian manifold with positive Ricci curvature, and let $S$ be a self-touching constant mean curvature (CMC) surface in $M$. Finiteness conjecture. The surface $S$ cannot have infinitely many touching points. This conjecture asks whether positive Ricci curvature restricts the touching behavior of CMC surfaces; the supplied text gives no evidence of a resolution.