The rainbow tree conjecture for properly coloured regular graphs

Let $G$ be a properly coloured $d$-regular graph, meaning that the edges incident with each vertex have distinct colours. A copy is rainbow if all its edges have distinct colours.

Rainbow tree conjecture. For every tree $T$ on $d$ vertices, every properly coloured $d$-regular graph $G$ contains a rainbow copy of $T$.

This conjecture generalises the rainbow path problem from complete graphs to regular graphs and trees. The paper leaves it as an open direction, while proving related results for Cayley-sum colourings.