Asymptotic enumeration conjecture for graphical regular representations

Let $d\text{-valent}$ vertex-transitive graphs, Cayley graphs, and graphical regular representations (GRRs) be counted up to order at most $n$ by $VT_d(n)$, $CAY_d(n)$, and $GRR_d(n)$, respectively. GRR enumeration conjecture. There exist positive constants $a$, $b$ and $c$ such that, for every $d\geq 3$,

$$n^{ad\log n}-c\leq GRR_d(n)\leq CAY_d(n)\leq VT_d(n)\leq n^{bd\log n}.$$

This extends the established bounds for valency $3$ to every fixed valency $d\geq 3$; the conjecture remains open.