  The Four Colour Theorem asserts that the vertices of every plane graph can be
properly coloured with four colors. Fabrici and G\"oring conjectured the
following stronger statement to also hold: the vertices of every plane graph
can be properly coloured with the numbers 1,...,4 in such a way that every face
contains a unique vertex coloured with the maximal color appearing on that
face. They proved that every plane graph has such a colouring with the numbers
1,...,6. We prove that every plane graph has such a colouring with the numbers
1,...,5 and we also prove the list variant of the statement for lists of sizes
seven.
