Gyárfás and Lehel [GyLe78] proved that this holds if all but at most $2$ of the trees are stars, or if all the trees are stars or paths. Fishburn [Fi83] proved this for $n\leq 9$. Bollobás [Bo83] proved that the smallest $\lfloor n/\sqrt{2}\rfloor$ many trees can always be packed greedily into $K_n$.

Joos, Kim, Kühn, and Osthus [JKKO19] proved that this conjecture holds when the trees have bounded maximum degree. Allen, Böttcher, Clemens, Hladky, Piguet, and Taraz [ABCHPT21] proved that this conjecture holds when all the trees have maximum degree $\leq c\frac{n}{\log n}$ for some constant $c>0$.

Janzer and Montgomery [JaMo24] have proved that there exists some $c>0$ such that the largest $cn$ trees can be packed into $K_n$.