Vanishing algebraic connectivity for regular graphs

For $d\geq 1$, let the maximum range over all $2d$-regular graphs $G$ on $n$ vertices be taken. Regular-graph vanishing conjecture.

$$\lim_{n\to\infty}\max\{a_d(G):\,G\text{ is $2d$-regular on $n$ vertices}\}=0.$$

This is presented as a possible generalization of a conjecture attributed in the source to Jordan. It is motivated by computer calculations, but the source gives no proof or resolution.