The periodic-structure conjecture for most frequent subwords

Let $w$ be a word, and write $w^m$ for its $m$-fold concatenation. A subword is most frequent if it maximizes the number of occurrences in $w^m$. Periodic-structure conjecture. There is a word $v$ such that, for every sufficiently large $m$, some most frequent subword of $w^m$ has the form

$$u \cdot v^r \cdot u',$$

where $u$ and $u'$ have lengths bounded by $|v|$.

If true, this would reduce the asymptotic analysis of most frequent subwords in periodic words to powers of a fixed word, potentially yielding improved bounds for the subword-entropy constants. The claim is supported by examples and is presented as open.