Conjecture that asymptotically minimum algebraic connectivity implies maximum diameter

Let $d\ge3$, and let $\Gamma_n$ be a sequence of graphs with minimum degree $\delta=d$, or a sequence of $d$-regular graphs. Say that the sequence has asymptotically minimum algebraic connectivity if its algebraic connectivity is asymptotically the smallest possible in the relevant class, and asymptotically maximum diameter if its diameter is asymptotically maximal.

Minimum-connectivity–maximum-diameter conjecture. If $\Gamma_n$ has asymptotically minimum algebraic connectivity, then its diameter is asymptotically maximum, namely

$$\operatorname{diam}(\Gamma_n)=(1+o(1))\frac{3n}{d+1}.$$

The paper proves that maximum diameter does not generally force asymptotically minimum algebraic connectivity for larger degrees, while proposing this converse direction as a general principle.