  A facial unique-maximum coloring of a plane graph is a proper vertex coloring
by natural numbers where on each face $\alpha$ the maximal color appears
exactly once on the vertices of $\alpha$. Fabrici and G\"oring proved that six
colors are enough for any plane graph and conjectured that four colors suffice.
This conjecture is a strengthening of the Four Color theorem. Wendland later
decreased the upper bound from six to five. In this note, we disprove the
conjecture by giving an infinite family of counterexamples. Thus we conclude
that facial unique-maximum chromatic number of the sphere is five.
