Hitting all maximum cliques with a stable set using lopsided independent transversals

Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a stable set meeting all maximum cliques. We strengthen this result, proving that such a stable set exists for any graph with omega > (2/3)(Delta+1). This is tight, i.e. the inequality in the statement must be strict. The proof relies on finding an independent transversal in a graph partitioned into vertex sets of unequal size.