Empirical asymptotics for extension-critical and prefix normal word counts

Let $\Sigma=\{0,1\}$, let $\mathcal L_{\mathrm{PN1}}$ be the language of prefix normal words, and let $\textit{ecrit}(n)$ count the extension-critical words in $\mathcal L_{\mathrm{PN1}}\cap\Sigma^n$, where an extension-critical word is a prefix normal word $w$ such that $w1$ is not prefix normal. Let $\textit{pnw}(n)$ denote the number of prefix normal words of length $n$. Extension-critical asymptotic conjecture. As $n$ tends to infinity,

$$\textit{ecrit}(n)=\textit{pnw}(n)\,\Theta\left(\frac{\log n}{n}\right),$$

and

$$\textit{pnw}(n)=2^{n-\Theta((\log n)^2)}.$$

These are empirical asymptotic predictions refining the conjectured convergence of the extension-critical ratio to zero; the paper presents numerical evidence but no proof.