Distinct prefix normal words have distinct extension-count sequences

Let $v,w e 0$ be distinct prefix normal binary words, with extension counts $\textit{ext}(v,m)$ and $\textit{ext}(w,m)$ denoting the numbers of prefix normal extensions of length $m$. Distinct-extension-sequence 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 the analogous claim for the extension languages themselves, but the paper does not prove that their growth sequences differ.