The finiteness conjecture for homoclinic classes of Venice masks

Let $X$ be a Venice mask on a compact manifold, and let $M(X)$ denote its maximal invariant set. A homoclinic class is the homoclinic class associated with a periodic orbit. The finiteness conjecture. The maximal invariant set of every Venice mask is finite union of homoclinic classes. The paper explains that the maximal invariant set is known to be a union of countably many homoclinic classes, and presents this finite-union statement as a consequence that would follow if the two preceding conjectures hold. Its resolution is not supplied, so the conjecture remains open.