  We prove that in any hyperbolic orbifold with one boundary component, the
product of any hyperbolic fundamental group element with a sufficiently large
multiple of the boundary is represented by a geodesic loop that virtually
bounds an immersed surface. In the case that the orbifold is a disk, there are
some conditions. Our results generalize work of Calegari-Louwsma and resolve a
conjecture of Calegari.
