Linear asymptotics conjecture for the Waiter-Client tree degree threshold

Let $D(n)$ be the largest integer $t$ such that for every tree $T$ with $v(T)=n$ and $\Delta(T)\leq t$, Waiter has a winning strategy in $\operatorname{WC}(n,T)$. Linear asymptotics conjecture. There exists a constant $c$ such that

$$D(n)=cn+o(n).$$

The paper establishes linear lower and upper bounds on $D(n)$ that differ by a multiplicative constant; the conjecture asserts that a limiting linear constant exists.