Conjecture on packing families of bounded-degree trees with non-spanning members

Let a tree family be a family $(T_s)_{s\in[N]}$ of trees. Bounded-degree tree-family packing conjecture. There exist $\delta>0$ and $n_0\in\mathbb{N}$ such that, for every $n\ge n_0$, every tree family $(T_s)_{s\in[N]}$ satisfying

$$\Delta(T_s)\le n/2\quad\text{and}\quad v(T_s)\le n\qquad\text{for all }s\in[N],$$ $$\delta n\le v(T_s)\le(1-\delta)n\qquad\text{for all }s\in[\delta n],$$

and

$$\sum_{s\in[N]}e(T_s)\le\binom{n}{2}$$

packs into $K_n$. This is motivated by necessary constraints on general tree families, including the need for many non-spanning trees; the paper does not report a resolution of this conjecture.