Monotonicity and 3-power denominators of the extremal complexity sequences

Let $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and $(c_\alpha)_{\alpha<\xi}$ be the transfinite sequences of rational numbers from the preceding conjecture. Denominator conjecture. The three sequences are decreasing, and the denominator of every term $a_\alpha$, $b_\alpha$, or $c_\alpha$ is a power of $3$.

This conjecture records the additional structure suggested by the displayed initial terms of the three sequences. The source presents it as an unproved conjecture based on the observed pattern.