Maker's bounded-degree tree-building conjecture

Let $K_n$ be the complete graph on $n$ vertices, and let ${\mathcal T}_n$ denote the family of spanning trees on these vertices. In the Maker–Breaker game $(E(K_n),{\mathcal T}_n)$, Maker claims edges while Breaker claims the remaining edges; write $\Delta(T)$ for the maximum degree of a tree $T$. Maker's bounded-degree tree-building conjecture. For every positive integer $\Delta$, there exists an integer $n_0=n_0(\Delta)$ such that, for every $n\geq n_0$ and every tree $T=(V,E)$ with $|V|=n$ and $\Delta(T)\leq\Delta$, Maker has a strategy to win the game $(E(K_n),{\mathcal T}_n)$ within $n$ moves. The conjecture asserts that every sufficiently large bounded-degree tree can be built with at most one wasted move beyond the theoretical minimum of $n-1$ moves; the paper establishes this bound for trees with a long bare path and an $n+1$-move bound in general, leaving the stated uniform $n$-move bound open.