Rainbow minimum-degree conjecture for hypercubes

Prove that for every tree T with k edges, the maximum minimum degree of a properly edge-colored subgraph of the n-dimensional hypercube Q_n containing no rainbow copy of T is exactly k-1.

Background

The paper introduces a rainbow analogue of a weaker form of the Erdős–Sós problem by defining (G,F) as the maximum minimum degree among subgraphs of G that admit a proper edge coloring with no rainbow copy of F.

A union of k-1 coordinate matchings in Q_n gives the lower bound (Q_n,T)≥k-1 for every k-edge tree. The conjecture asks whether this lower bound is always sharp, independently of the structure of T. The authors prove it for paths, trees with sufficiently long pendant paths, trees with many leaves, and specified classes of spiders, but not for all trees.