Hajós' 4-coloring conjecture for graphs without a topological $K_5$

Let $G$ be a graph, and let $TK_5$ denote a subdivision of $K_5$. A graph is 4-colorable if its vertices can be colored with four colors so that adjacent vertices receive different colors.

Hajós' conjecture. Every graph containing no $TK_5$ is 4-colorable.

The conjecture would generalize the Four Color Theorem. The supplied source does not give a resolution for the $k=4$ case, so the conjecture remains open.