The asymptotic anti-power conjecture for shifted Thue–Morse words

Let $\gamma_j(k)$ be the function measuring the least relevant anti-power length of order $k$ for the shift parameter $j$ in the Thue–Morse word. For each nonnegative integer $j$, consider the lower and upper limiting ratios of $\gamma_j(k)$ to $k$. Asymptotic anti-power conjecture. For any nonnegative integer $j$,

$$\liminf_{k\rightarrow\infty}\frac{\gamma_j(k)}{k}=\frac{9}{10}\qquad\text{and}\qquad\limsup_{k\rightarrow\infty}\frac{\gamma_j(k)}{k}=\frac{3}{2}.$$

The conjecture extends Defant's corresponding conjecture for $j=0$; the paper establishes only bounds for these limits and leaves the exact asymptotic behavior open.