Claw-free graphs, skeletal graphs, and a stronger conjecture on $\omega$, $\Delta$, and $\chi$

The second author's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisties $\chi \leq \lceil \frac 12 (\Delta+1+\omega)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results we introduce a very useful $\chi$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called "skeletal" graphs.