Asymptotic tightness of the lower bound for 1-perfect binary codes

Let $B(n-1)$ denote the number of $1$-perfect binary codes of length $n-1=2^m-1$, and let $\widetilde K_{\scriptscriptstyle LA}(n)$ be the lower-bound expression satisfying $B(n-1)\geq \widetilde K_{\scriptscriptstyle LA}(n)$. The expression in equation (4) is the asymptotic formula for $\widetilde K_{\scriptscriptstyle LA}(n)$. Asymptotic tightness conjecture. The lower bound is asymptotically tight: equation (4) is the asymptotic number of $1$-perfect binary codes of length $n-1=2^m-1$. This would determine the asymptotic growth of the number of $1$-perfect binary codes, beyond the established constructive lower bound; no resolution is supplied in the source.