Nearly platonic graph two-exceptional-faces conjecture

Let $G$ be a vertex-regular planar graph, and suppose that all but two faces of $G$ have the same degree. Nearly platonic graph conjecture. The remaining two faces must have the same degree as each other. The conjecture arises from the construction of nearly platonic graphs, in which the two exceptional faces have consistently been observed to have equal degree. The authors report that they have been unable to find a counterexample.