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

Let $b$ be the Thue–Morse word, and let $AP_j(\mathbf{t},k)$ denote the set of lengths of anti-power $j$-fixes of order $k$ in $b$. For fixed $j,k\in\mathbb{Z}^{\geq 0}$ with $k\geq 3$, consider the membership of a positive integer $m$ and its double $2m$. Anti-power doubling conjecture. For any fixed $j,k\in\mathbb{Z}^{\geq 0}$ with $k\geq 3$, the equivalence

$$m\in AP_j(\mathbf{t},k)\iff 2m\in AP_j(\mathbf{t},k)$$

holds for all but finitely many $m\in\mathbb{Z}^+$. This would justify focusing on odd lengths when defining $\Gamma_j(k)$, although the assertion remains unproved.