Linear χ‑boundedness of complements of h‑perfect graphs

Prove that the class of complements of h‑perfect graphs is linearly χ‑bounded; equivalently, develop a linear function f such that for every graph G in the class and every induced subgraph H of G, χ(H) ≤ f(ω(H)).

Background

The authors show that complements of h‑perfect graphs are quadratically χ‑bounded, giving χ(G) ≤ (ω(G)+1 choose 2). They conjecture that this bound can be improved to a linear one, paralleling linear χ‑boundedness known for perfect graphs and extending recent χ‑boundedness developments to this polyhedrally defined class.