The vanishing proportion conjecture for trees attaining the Laplacian lower bound

Let $T$ be a tree of order $n$, let $d$ denote its diameter, and let $m_T[0,1)$ be the number of Laplacian eigenvalues of $T$ in the interval $[0,1)$. Define

$$\mathcal{T}_{n}^{*}=\{T\in\mathcal{T}_{n}:m_T[0,1)=\lceil (d+1)/3\rceil\},$$

where $\mathcal{T}_{n}$ is the set of all trees of order $n$. Vanishing proportion conjecture.

$$\lim_{n\to\infty}\frac{\#\mathcal{T}_{n}^{*}}{\#\mathcal{T}_{n}}=0.$$

The conjecture asserts that trees attaining the general lower bound for the number of Laplacian eigenvalues below $1$ become negligible among all trees of order $n$. It is based on computations for trees with at most $20$ vertices and is presented as motivation for improving the lower bound for almost all trees.