Asymptotic formula for flip-graph vertices of repeated even chains

Let $\ell$ be a fixed even positive integer, and let $G(\underbrace{\ell,\dots,\ell}_{n\text{ copies}})$ denote the flip graph associated with the product of $n$ chains of length $\ell$. Asymptotic vertex-count conjecture. As $n$ tends to infinity,

$$\log v(G(\underbrace{\ell,\dots,\ell}_{n\text{ copies}})) \sim C\frac{(2\ell)^n}{\sqrt{n}},$$

where

$$C=\frac{\log 2}{\sqrt{2\pi(2\ell-1)(2\ell+1)/3}}.$$

The paper gives exact asymptotics in dimensions two and three and notes that the exact logarithmic asymptotic is open for dimensions at least four; this formula is proposed for the repeated-chain family.