Haythorpe's lower-bound conjecture for Hamiltonian cycles in regular graphs

Let $r$ and $n$ be integers with $r\geq 5$ and $n\geq r+3$, and let $G$ be a Hamiltonian $r$-regular graph on $n$ vertices. Haythorpe's conjecture. The graph $G$ has at least

$$(r-1)^2 ((r-2)!)^{\frac{n}{r+1}}$$

Hamiltonian cycles. The conjecture proposes an asymptotic lower bound matching the construction discussed in the source, while the paper explains that its upper-bound results disprove the conjecture in a strong asymptotic sense.