The Cantor-normal-form conjecture for logics of ordinal beta spaces

Let $\gamma$ be an ordinal with Cantor normal form, and write $\alpha_1$ for its leading exponent. Let $\beta(\gamma)$ be the corresponding beta space, $sf L(\beta(\gamma))$ its logic, and $sf L_\infty=\sf L(\mathcal R_\infty)$ the logic of all roaches. Cantor-normal-form conjecture. The following two assertions hold:

1. If $\alpha_1\ge\omega$, then

$${\sf L}(\beta(\gamma))={\sf L}_\infty={\sf L}(\mathcal R_\infty).$$

2. The list of logics ${\sf L}(\beta(\gamma))$ arising from ordinals $\gamma$ is obtained by adding ${\sf L}_\infty$ to the list in the cited partial solution of Shehtman's second problem.

The paper gives a partial solution for ordinals of a special Cantor-normal-form type; the arbitrary-ordinal case remains open.