Periodicity and Thue–Morse structure of the sequences associated with $B_a$ and $C_a$

Let $B_a$ and $C_a$ be the sets of nonnegative integers satisfying, respectively, $t(x+a)=1-t(x)$ and $t(x+a)=t(x)$, where $t(x)$ is the Thue–Morse sequence. Let $\{B_a(n)\}$ and $\{C_a(n)\}$ list these sets in increasing order, and let $\{\beta_a(n)\}$ and $\{\gamma_a(n)\}$ be the binary sequences obtained by recording whether the listed integers are odious or evil. An integer is evil if its binary expansion contains an even number of $1$'s and odious if it contains an odd number of $1$'s.

Periodicity conjecture. The sequence $\{\gamma_a(n)\}$ is periodic. If $2^m\mathrel{\|}a$, then its minimal period has $2^{m+1}$ terms. If $a$ is evil, this period consists of the first $2^{m+1}$ terms of the Thue–Morse sequence $\{t_n\}$; otherwise, it consists of the first $2^{m+1}$ terms of $\{1-t_n\}$. Moreover, $\beta_a(n)+\gamma_a(n)=1$.

The claim describes a precise periodic structure for the solutions of the two complementary Thue–Morse equations and gives an explicit relation between their parity-coded enumerations. The supplied text does not indicate whether the conjecture has been resolved.