Betweenness center of a uniformly random tree

Prove or disprove that, for a uniformly random unlabeled tree T_n of order n, the probability that the betweenness center has order one converges to one as n tends to infinity.

Background

The paper’s enumeration of trees through order 20 shows that betweenness centers are typically small in the examined range, while the preceding discussion notes that trees with arbitrarily large centers nevertheless exist. This motivates a probabilistic conjecture concerning the asymptotic behavior of the betweenness center of a uniformly random unlabeled tree.

The conjecture specifically predicts that almost every sufficiently large unlabeled tree has a unique vertex of maximum betweenness.