Weight of edges in normal plane maps

The purpose of our note is to prove that every NPM has either a simplicial 3-vertex adjacent to a vertex degree at most 10, or (3,9)-edge, or (4,7)-edge, or (5,6)-edge. In particular, this confirms the above mentioned conjecture by Ferencová and Madaras. Furthermore, we construct a 3-polytope showing that the above term (3,9) is best possible.