Chromatic graph generalization of the Tree Packing Conjecture

Let $T_2,\dots,T_k$ be trees, where $T_i$ has $i$ vertices, and let a packing mean edge-disjoint copies of the trees in a host graph.

Chromatic tree-packing conjecture. If $G$ is a $k$-chromatic graph, then the set of trees $T_2,\dots,T_k$ has a packing into $G$.

This conjecture would imply the Tree Packing Conjecture, since the complete graph on $k$ vertices is $k$-chromatic. The paper develops results for special families of trees and chromatic graphs, but the full statement remains open.