Transfinite sequences describing the greatest numbers of each complexity class modulo 3

Let $\|n\|$ denote the complexity of a natural number, and let $\xi$ be an ordinal. Consider three transfinite sequences $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and $(c_\alpha)_{\alpha<\xi}$ of rational numbers. Transfinite-sequence conjecture. There exist such sequences for which the greatest numbers of complexity $3n$, respectively $3n+1$ and $3n+2$, are the first natural numbers contained in $(3^n a_\alpha)$, respectively $(3^n b_\alpha)$ and $(3^n c_\alpha)$, and $\xi$ is an infinite countable ordinal satisfying

$$\omega\xi=\xi.$$

The conjecture proposes a transfinite organization of the extremal natural numbers by complexity modulo $3$. The paper gives initial numerical data but no proof or resolution.