Conjecture on minimizers of the number of Lyndon factors

Let $\ell(n)$ be the minimum number of Lyndon factors among Lyndon words of length $n$:

$$\ell(n)=\min\{\mathcal{L}(w)\mid w\text{ is a Lyndon word with }|w|=n\}.$$

A minimizer conjecture. If $w$ is a Lyndon word with $|w|\neq 6$ and $\mathcal{L}(w)=\ell(|w|)$, then $w$ is a Sturmian Lyndon word, meaning that there exist distinct letters $a,b$ such that

$$w\in\{a,b\}^{+},\qquad w=ap_wb,$$

where $p_w$ is a central word.

This conjecture characterizes the Lyndon words attaining the minimum possible number of Lyndon factors at each length, apart from the exceptional length $6$. It extends the preceding observation that Fibonacci Lyndon words are optimal and addresses the existence of optimal Lyndon words using more than two letters; the source does not state whether the conjecture has been resolved.