Conjecture on Thue–Morse maximum order complexity along polynomial subsequences

Let $\mathcal{T}_P$ denote the Thue–Morse sequence along a polynomial subsequence, where $P$ is a polynomial of degree $d$. Write $M(\mathcal{T}_P,N)$ for its maximum order complexity up to length $N$. Thue–Morse polynomial-subsequence conjecture. There are constants $c,C>0$ such that, for all sufficiently large $N$,

$$cN^{1/d}\leq M(\mathcal{T}_P,N)\leq CN^{1/d}.$$

Equivalently, $M(\mathcal{T}_P,N)\asymp N^{1/d}$. A proof would show that the previously proved lower bound is optimal; the claim is supported by computations for squares and cubes.