The asymptotic cardinality conjecture for Wiener indices of trees

Let $\mathcal{T}_{n}$ denote the class of trees on $n$ vertices, and let $W[\mathcal{T}_{n}]$ be the set of their Wiener indices. Cardinality conjecture. The cardinality of $W[\mathcal{T}_{n}]$ equals

$$\frac{1}{6}n^{3}+O(n^{2}).$$

This is one of the conjectures concerning the Wiener inverse interval problem for trees; it predicts the asymptotic number of distinct Wiener-index values realized by trees of order $n$.