Conjecture on the maximum length of rich square-free words

Let $r(n)$ be the maximum length of a rich square-free word over an alphabet of size $n$, and let $w_n$ be the recursively constructed rich square-free word over an alphabet of size $n$. Conjecture. For every $n\geq 1$,

$$r(n)=\operatorname{max}\\{|w_n|,2\cdot|w_{n-1}|+1\\}.$$

This conjecture asserts that the maximum length is the larger of the length of the paper's recursively constructed word and the length obtained from the basic recursion. The computed values for $n=7,8,9,10$ motivate the conjecture, but the exact values for $n=8$ and $n=9$ were not computable in the paper.