Radius three trees in graphs with large chromatic number

A class $\Gamma$ of graphs is $\chi$-bounded if there exists a function $f$ such that $\chi \left(G\right) \leq f \left(\omega \left(G\right) \right)$ for all graphs $G \in \Gamma$, where $\chi$ denotes chromatic number and $\omega$ denotes clique number. Gyarfas and Sumner independently conjectured that, for any tree T, the class ${\rm Forb} \left(T\right)$, consisting of graphs that do not contain T as an induced subgraph, is $\chi$-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.