Limit relations between successive blocks of the transfinite complexity sequences

Let $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and $(c_\alpha)_{\alpha<\xi}$ be the transfinite sequences associated with the three complexity classes modulo $3$. Block-limit conjecture. For all ordinals $\beta<\xi$,

$$\lim_{n\to\infty}a_{\omega\beta+n}=\frac{c_\beta}{3},\qquad \lim_{n\to\infty}b_{\omega\beta+n}=a_\beta,\qquad \lim_{n\to\infty}c_{\omega\beta+n}=b_\beta.$$

The source explicitly describes these conjectures as more doubtful and based on only a few cases. They are used to motivate the possible size of $\xi$, including the observation that $\omega^\omega$ is the least solution of $\omega\xi=\xi$.