Waiter–Client tree-universality conjecture

Let $K_n$ be the complete graph on $n$ vertices. A graph is tree-universal at degree bound $D$ if it contains a copy of every tree $T$ with $n$ vertices and maximum degree $\Delta(T)\leq D$. Waiter–Client tree-universality conjecture. There exists a constant $c>0$ such that, for every sufficiently large integer $n$, in the $(1:1)$ Waiter–Client game on $K_n$, Waiter has a strategy forcing Client to claim a graph that is tree-universal at degree bound $cn$. The source says that the established degree order in the Waiter–Client theorem should be improvable, but gives no resolution of this conjecture.