Limit conjecture for repetition thresholds of rich words

For each integer $k\geq 2$, let $R_k$ denote the language of infinite rich words over a $k$-letter alphabet, and let $\operatorname{RT}(R_k)$ be its repetition threshold.

Limit conjecture. The repetition thresholds of rich words satisfy

$$\lim_{k\to\infty}\operatorname{RT}(R_k)=2.$$

The known values for $k=2$ and $k=3$, together with computational evidence for larger alphabets, motivate this asymptotic prediction; its general proof remains open.