Strong boundary-word existence conjecture

For each integer $n\ge 5$, let ${\bf A}_n$ be the $n$-letter alphabet, ${\bf A}_n^{\omega}$ the set of infinite words over it, ${\bf T}_n$ the class of boundary words, and let ${\bf F}(\mathbf w)$ denote the set of factors of an infinite word $\mathbf w$. For a factor $\mathbf x$, let ${\sf \exp}(\mathbf x)$ be its exponent. Strong boundary-word existence conjecture. For every $n\ge5$, there exists an infinite word $\mathbf w\in{\bf T}_n\cap{\bf A}_n^{\omega}$ such that

$$\max_{\substack{\mathbf x\in{\bf F}(\mathbf w),\ |\mathbf x|\ge l}}{\sf \exp}(\mathbf x)\xrightarrow{l\to\infty}1.$$

This is the main hypothesis introduced in this part of the paper: it asserts the existence, for every alphabet with at least five letters, of a boundary word whose sufficiently long factors have exponents arbitrarily close to $1$. The surrounding text indicates that an appropriately weakened version is proved from the exponential hypothesis, while the full assertion is presented as a conjecture.