The algebraic-connectivity conjecture for complete graphs in $\ell_\infty^d$

Let $a(G,X)$ denote the algebraic connectivity of a graph $G$ in a normed space $X$, let $K_{2d}$ be the complete graph on $2d$ vertices, and let $T_d$ be the tree defined in the paper. Algebraic-connectivity conjecture.

$$a(K_{2d},\ell_\infty^d)=a(T_d)\quad\text{for }d\geq 4.$$

This conjecture compares the algebraic connectivity of the complete graph in the $d$-dimensional $\ell_\infty$ space with that of the specified tree $T_d$; the supplied excerpt gives no resolution status.