Rainbow Erdős–Sós conjecture for hypercubes

Prove that for every positive integer n and every tree T with k edges, the relative rainbow extremal number in the n-dimensional hypercube satisfies ex^*(Q_n,T)=ex^*(Q_n,K_{1,k}).

Background

The n-dimensional hypercube Q_n has a natural 1-factorization into coordinate matchings. Since the union of k-1 coordinate matchings provides a properly colored subgraph avoiding every k-edge tree, the quantity ex^*(Q_n,T) has a common natural lower bound for all trees with k edges.

The conjecture asserts that this lower bound is optimal and that, within Q_n, the relative rainbow extremal number depends only on the size of the forbidden tree. The paper verifies the claim for paths of lengths three and four and for an infinite family of trees with many leaves, while establishing only a weaker general upper bound for arbitrary trees.