The reduced-denominator multiplicity conjecture for hard squares on cylinders

Let $P_m$ be the path graph on $m$ vertices and $C_n$ the cycle graph on $n$ vertices. Define

$$f_n(t)=\sum_{m=0}^{\infty} Z(P_m\times C_n)t^m,$$

where $Z(G)=1-\chi(\operatorname{Ind}(G))$ is the Witten index associated with the independence complex of $G$. Write each rational function $f_n(t)$ after cancelling common factors between its numerator and denominator. Reduced-denominator multiplicity conjecture. After this reduction, $f_{4n+2}(t)$ has a denominator with no multiple zeroes, and $f_{4n}(t)$ has a denominator whose zeroes have multiplicity at most two. Consequently, for every fixed $n$, the sequence $a_m=Z(P_m\times C_{4n+2})$ is periodic, while the sequence $a_m=Z(P_m\times C_{4n})$ has linear growth. This is a proposed refinement of the established rationality and root-of-unity result for the generating functions; the asserted denominator multiplicity and resulting periodicity or linear-growth conclusions remain open in the source.