The rational generating-function denominator 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. For a graph $G$, let $\operatorname{Ind}(G)$ be its independence complex, define $Z(G)=1-\chi(\operatorname{Ind}(G))$, and set

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

Generating-function denominator conjecture. For every $n\geq 0$, there are polynomials $h_n(t)$ such that

$$f_{4n+2}(t)=\frac{h_{4n+2}(t)}{(1+t^2)\cdot\big[(1-t^{8n-2})(1-t^{8n-8})(1-t^{8n-14})\cdots(1-t^{2n+4})\big]},$$ $$f_{4n}(t)=\frac{h_{4n}(t)}{(1-t^2)\cdot\big[(1-t^{8n-6})(1-t^{8n-12})(1-t^{8n-18})\cdots(1-t^{2n+6})\big]}.$$

Here, in each denominator, the displayed exponents decrease by $6$. The preceding theorem establishes that each $f_{2n}(t)$ is rational with denominator roots of unity; this conjecture predicts explicit denominator factors and would give more precise polynomial-growth information for the Witten-index sequences of hard-square models on even-circumference cylinders.