Unique-maximum edge-colouring of plane graphs with respect to faces

A unique-maximum kk-edge-colouring with respect to faces of a 2-edge-connected plane graph GG is an edge-colouring with colours 1,…,k1,…,k so that, for each face αα of GG, the maximum colour occurs exactly once on the edges of αα. We prove that any 2-edge-connected plane graph has such a colouring with 3 colours. If we require the colouring to be facially proper then 6 colours are enough.