Defant's asymptotic conjecture for odd antipower prefix lengths

Let $\mathbf{t}$ be the Thue–Morse word, and for each positive integer $k$ let $\gamma(k)$ be the minimum odd positive integer $n$ such that the prefix of $\mathbf{t}$ of length $kn$ is a $k$-antipower. Here, a word of length $kn$ is a $k$-antipower if it can be written as $w_1\cdots w_k$, where each $w_i$ has length $n$ and $w_i\ne w_j$ for $i\ne j$.

Defant's conjecture.

$$\liminf\limits_{k \to \infty} \frac{\gamma(k)}{k} = \frac{9}{10}$$

\nand

$$\limsup\limits_{k \to \infty} \frac{\gamma(k)}{k} = \frac{3}{2}.$$

These equalities would sharpen the known bounds on the asymptotic growth of $\gamma(k)$, complementing the established exact values for the corresponding lower and upper limits of $\Gamma(k)/k$.