Conjecture on Zeckendorf sum-of-digits maximum order complexity along polynomial subsequences

Let $s_Z(n)$ denote the sum of digits of $n$ in Zeckendorf base, let $\mathcal{S}_Z=(s_Z(n)\bmod 2)_n$, and let $\mathcal{S}_{Z,P}$ be the sequence obtained by restricting $\mathcal{S}_Z$ to a polynomial subsequence $P(n)$, where $P$ has degree $d\geq2$. Write $M(\mathcal{S}_{Z,P},N)$ for its maximum order complexity up to length $N$. Zeckendorf polynomial-subsequence conjecture. There are constants $c,C>0$ such that, for all sufficiently large $N$,

$$cN^{1/(2d)}\leq M(\mathcal{S}_{Z,P},N)\leq CN^{1/(2d)}.$$

Equivalently, $M(\mathcal{S}_{Z,P},N)\asymp N^{1/(2d)}$. The paper proves the corresponding lower bound, so the conjecture asserts its sharpness; computational evidence is given for squares.