The fiberedness conjecture for cyclic tilings

Let $A\oplus B=\mathbb{Z}_M$ be a tiling, where $M=\prod_{j=1}^K p_j^{n_j}$. Say that a set is $M/p_i^{\alpha_i}$-fibered in the $p_i$ direction when it has the corresponding fibered structure on that scale.

Fiberedness conjecture. For every $i$ there exists $1\leq\alpha_i<n_i$ such that either $A$ or $B$ is $M/p_i^{\alpha_i}$-fibered in the $p_i$ direction. In particular, if $M$ has $3$ prime factors, $\Phi_M\mid A$, and there exists a $D(M)$ grid $\Lambda$ such that $A\cap\Lambda$ is not fibered in any direction, then $B$ is fibered in all directions on some scale.

The conjecture is motivated by the role of fibering in the paper's tiling arguments and by the behavior of Szabó-type examples. The source presents it as an unresolved structural assertion.