Conjecture on the asymptotic ratios for anti-power prefixes of the Thue–Morse word

Let ${\bf t}$ be the Thue–Morse word, and let $\gamma(k)$ denote the least positive integer $n$ such that the prefix of ${\bf t}$ of length $nk$ is a $k$-anti-power. Then the asymptotic-ratio conjecture.

$$\liminf_{k\to\infty}\frac{\gamma(k)}{k}=\frac{9}{10}\hspace{.5cm}\text{and}\hspace{.5cm}\limsup_{k\to\infty}\frac{\gamma(k)}{k}=\frac{3}{2}.$$

The paper establishes upper bounds consistent with these values and notes that numerical plots suggest they are sharp; determining the exact values was left unresolved in the source.