Extremality conjecture for generalized path graphs

Let $d\geq 1$ and let $P_{n,d}$ be the generalized path graph, an $n$-vertex minimally $d$-rigid graph. For a graph $G$, let $a_d(G)$ denote its $d$-dimensional algebraic connectivity. Extremality conjecture for generalized path graphs. If $G$ is a $d$-rigid graph on $n$ vertices, then

$$a_d(G)\geq a_d(P_{n,d}).$$

The generalized path graphs satisfy $a_d(P_{n,d})=\Theta_d(1/n^2)$, so the conjecture proposes that they minimize $d$-dimensional algebraic connectivity among $n$-vertex $d$-rigid graphs. Its resolution is not given in the source.