Average-degree generalization of the Tree Packing Conjecture

Let $T_2,\dots,T_k$ be trees, where $T_i$ has $i$ vertices, and let $G$ be a graph on $n$ vertices.

Average-degree tree-packing conjecture. If $G$ has average degree at least $k-1$, equivalently at least

$$\frac{k-1}{2}n$$

edges, then the set of trees $T_2,\dots,T_k$ has a packing into $G$.

This strengthens the minimum-degree version by replacing minimum degree with average degree. The paper proves an analogue for packing only the smaller trees $T_2,\dots,T_s$ when $s\leq k/2$, but the full conjecture remains open.