Constant-error rainbow tree embedding conjecture

A properly coloured graph is an edge-coloured graph in which edges sharing a vertex have distinct colours. A subgraph is rainbow if all its edges have distinct colours. Let $K_n$ be the complete graph on $n$ vertices.

Constant-error rainbow tree conjecture. There is a constant $C$ such that every properly coloured $K_n$ has a rainbow copy of every tree on $n-C$ vertices.

The paper proves rainbow copies for every tree on $n-o(n)$ vertices and explains that the error term cannot be zero. The conjecture asks whether the error can be reduced to a constant.