Graphs that contain a $K_{1,2,3}$ and no induced subdivision of $K_4$ are $4$-colorable

In 2012, Lévêque, Maffray, and Trotignon conjectured that each graph $G$ that contains no induced subdivision of $K_4$ is $4$-colorable. In this paper, we prove that this conjecture holds when $G$ contains a $K_{1,2,3}$.