The bound on lengths of perfectly clustering words in Gessel's map

Let $n\geqslant 2$ and let $(\alpha_2,\ldots,\alpha_n)$ be an $(n-1)$-tuple of nonnegative integers. Let $\Phi$ be the map from words to multisets of conjugacy classes of primitive words described above. Length bound conjecture. The number of distinct lengths of conjugacy classes of words appearing in $\Phi(n^{\alpha_n}\dots 2^{\alpha_2})$ is at most

$$\left\lceil\frac{n-1}{2}\right\rceil.$$

This conjecture concerns the possible lengths of the conjugacy classes arising from the map $\Phi$ and was expressed in the second author's thesis. Its resolution is not specified in the source.