Colouring Squares of Claw-free Graphs

Is there some absolute epsilon > 0 such that for any claw-free graph G, the chromatic number of the square of G satisfies chi(G(2)) <= (2 - epsilon)omega(G)(2), where omega (G) is the clique number of G? Erdos and Nesetril asked this question for the specific case where G is the line graph of a simple graph, and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdos and Nesetril.