Perfect divisibility of E-free graphs

Prove that every E-free graph is perfectly divisible, where E=S_{1,2,2} is the tree formed by three paths of lengths 1, 2, and 2 sharing one common end.

Background

The paper defines S_{a,b,c} as the tree formed by three paths of lengths a, b, and c with one common end and otherwise disjoint vertices. The fork treated in the main theorem is S_{1,1,2}, while E=S_{1,2,2} is obtained from a claw by subdividing two different edges once.

The authors construct counterexamples showing that forbidding K_{1,4}, S_{1,1,3}, or S_{2,2,2} does not guarantee perfect divisibility. They explicitly state that these examples do not settle the corresponding E-free case and formulate the following conjecture.