The asymptotically optimal algebraic-connectivity bound for bounded-degree trees

Let $T$ be a tree with $n$ vertices and maximum degree $d$. The asymptotically optimal tree bound conjecture. As $n\rightarrow\infty$ for fixed $d$,

$$\lambda_{2}(T)\leq\frac{d(d-2)}{d-1}\frac{1}{n}+O\left(\frac{\ln n}{n^{2}}\right).$$

Here $\lambda_{2}(T)$ denotes the algebraic connectivity of $T$. The conjecture improves the paper's preceding general bound and is known in the source for the well-balanced Bethe trees; its validity for all trees with maximum degree $d$ remains open.