The almost-all-trees lower-bound conjecture for small Laplacian eigenvalues

Let $T$ be a tree with diameter $d$, and let $m_T[0,1)$ denote the number of Laplacian eigenvalues of $T$ less than $1$. Almost-all-trees lower-bound conjecture. Almost all trees have at least

$$\left\lceil\frac{d+1}{3}\right\rceil+1$$

Laplacian eigenvalues less than $1$. This would improve the general lower bound by one for asymptotically almost all trees. The statement is presented as a consequence of the conjecture that the proportion of trees attaining the existing lower bound tends to zero; the supplied text also says that this conjecture was confirmed independently by Sin and by Jacobs, Oliveira and Trevisan.