The three-prime-factor reduction conjecture for cyclic tilings

Let $M$ be a positive integer, and consider the tiling questions referred to in the source as Questions $q$-slab and $q$-obstruction.

Three-prime-factor reduction conjecture. The answers to Questions $q$-slab (both versions) and $q$-obstruction are affirmative when $M$ has at most $3$ distinct prime factors.

The conjecture concerns structural reductions for tilings of cyclic groups. The source notes that the corresponding answer is affirmative for $M=p_1^2p_2^2p_3^2$, while the assertion for all $M$ with at most three distinct prime factors is presented as conjectural.