The well-balanced Bethe tree extremal conjecture

Let $T$ be a tree with

$$n=\frac{d(d-1)^{K}-2}{d-2}$$

vertices and maximum degree $d$. The well-balanced Bethe tree extremal conjecture. Its algebraic connectivity is less than the algebraic connectivity of the well-balanced Bethe tree with $n$ vertices whose non-leaf vertices have degree $d$. This is presented as a stronger version of the preceding asymptotic bound conjecture. The source states that the preceding conjecture is true for the well-balanced Bethe trees, but does not report a proof of this stronger assertion for every such tree.