The n-roach conjecture for Shehtman's first problem

For each $n$, let ${\sf L}_n$ denote the logic associated with $\beta(\omega^n)$, and let ${\sf L}(\mathcal R_n)$ denote the logic of the class of $n$-roaches. n-roach conjecture. For $n\ge 3$, one has

$${\sf L}_n={\sf L}(\mathcal R_n).$$

The paper proves the corresponding case $n=2$ (using the continuum hypothesis), while a solution for $n\ge 3$ remains open.