Host graphs supporting a rainbow Erdős–Sós principle

Determine which host graphs G have a relative rainbow extremal number ex^*(G,T) that depends only on the number k of edges of a tree T, equivalently satisfying ex^*(G,T)=ex^*(G,K_{1,k})=ex(G,K_{1,k}) for every k-edge tree T and every k.

Background

For graphs G and F, the relative rainbow extremal number ex^*(G,F) is the maximum number of edges in a subgraph of G admitting a proper edge coloring with no rainbow copy of F. The authors observe that replacing the complete host K_n by an arbitrary host is necessary because the ordinary rainbow extremal number ex^*(n,T) can vary among trees with the same number of edges.

The question asks for a characterization of host graphs on which the rainbow extremal number becomes structure-independent for trees. The paper later proposes the hypercube as a candidate host, but does not resolve the question for general G.