Conjecture on the growth of the Thue–Morse antipower threshold

Let $\mathbf{t}$ be the Thue–Morse word. For an odd positive integer $n$, let $\mathfrak{K}(n)$ be the smallest positive integer $k$ such that the prefix of $\mathbf{t}$ of length $kn$ is not a $k$-antipower.

Growth conjecture. There exists a sequence $(a_i)$ converging to $0$ such that, whenever $n$ lies between $3\cdot 2^i$ and $2^{i+1}(1-a_i)$,

$$\mathfrak{K}(n) \le \mathfrak{K}(3\cdot 2^i+1).$$

The conjecture concerns the poorly understood intermediate region in the growth of $\mathfrak{K}(n)$ and would help clarify the asymptotic behavior of $\Gamma(k)-\gamma(k)$.