The short most frequent subword conjecture

Let $w$ be a word of length at least $2$. A subword is most frequent if it maximizes the number of occurrences in $w$. Short-subword conjecture. At least one most frequent subword of $w$ has length at most

$$\left\lceil \frac{|w|}{2} \right\rceil.$$

The claim is motivated by lower bounds on maximum subword occurrence counts and by the intuition that sufficiently long subwords cannot be optimal. It is presented as open.