Chromatic decrease under soft Barycentric refinement in dimension two

Construct a 2-manifold G satisfying c(\phi(G))<c(G), where \phi(G) denotes the soft Barycentric refinement of G.

Background

The paper compares the chromatic number of a manifold with that of its soft Barycentric refinement. It gives examples in higher dimensions where refinement lowers the chromatic number, but reports that no corresponding 2-manifold example is known. Finding such an example would also settle the Albertson–Stromquist conjecture mentioned in the same caption.