Minimax degrees of quasiplane graphs without $4$-faces

The $M$-degree of an edge $xy$ in a graph is the maximum of the degrees of $x$ and $y$. The minimax degree of a graph $G$ is the minimum over $M$-degrees of its edges. In order to get upper bounds on the game chromatic number, W. He et al showed that every planar graph $G$ without leaves and $4$-cycles has minimax degree at most $8$. This was improved by Borodin et al to the best possible bound $7$. Answering a question by D. West, we show that every plane graph $G$ without leaves and $4$-faces has minimax degree at most $15$. The bound is sharp. Similar results are obtained for graphs embeddable on the projective plane, torus and Klein bottle.