Adjacency perfect state transfer conjecture for trees

Let $T$ be a tree, let $P_2$ and $P_3$ denote the paths on two and three vertices, and say that $T$ has adjacency perfect state transfer if its adjacency-matrix continuous-time quantum walk transfers a state between two vertices with fidelity equal to $1$. Adjacency perfect state transfer conjecture. No tree except for $P_2$ and $P_3$ admits adjacency perfect state transfer. The paper establishes this for trees with a perfect matching and reports computational verification for all trees up to ten vertices, but leaves the general case open.