Extension-language separation conjecture for prefix normal words

Let $v,w\in 1\{0,1\}^*$ be distinct prefix normal binary words. For each $m\geq 1$, let $\textit{ext}(v,m)$ and $\textit{ext}(w,m)$ denote the numbers of length-$m$ extensions that produce prefix normal words. Extension-language separation conjecture. If $v\neq w$, then the infinite sequences

$$(\textit{ext}(v,m))_{m\geq 1}\quad\text{and}\quad(\textit{ext}(w,m))_{m\geq 1}$$

are different. The preceding lemma establishes separation of the extension languages themselves; whether their growth, or equivalently these counting sequences, always differs was left unresolved.