The Wiener inverse interval conjecture for 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. Define $W^{\int}[\mathcal{T}_{n}]$ to be the largest interval of contiguous integers contained in $W[\mathcal{T}_{n}]$. Wiener inverse interval conjecture. The cardinality of $W^{\int}[\mathcal{T}_{n}]$ equals $O(n^{3})$.

This conjecture concerns the length of the largest contiguous interval of Wiener-index values realized by trees with $n$ vertices. It is presented as one of the conjectures underlying the Wiener inverse interval problem for trees.