Cyclic sieving conjecture for affine bicolor dichotomy patterns

Let $k$ be a positive integer, let $G_N=\operatorname{Aff}(\mathbb{Z}_{2k})$, and let $\{G_i\}$ be representatives of the orbits of the conjugation action, ordered so that $|G_N|\geq\cdots\geq|G_1|$. Let $B=(b_{i,j})$ be the inverse of the table of marks, and let $P_j$ denote the relevant pattern inventory polynomials. For each subgroup $G_i$, define

$$Q_i=\sum_{j=1}^{N}b_{i,j}P_j(1+x,1+x^2,\ldots,1+x^{|D|}).$$

Cyclic sieving conjecture. If $k$ is equal to $1$, $2$, or a power of an odd prime, then $|Q_i(-1)|$ counts the number of self-complementary dichotomies with automorphism group $G_i$. In particular, $|Q_1(-1)|$ counts the number of strong dichotomies.

The conjecture proposes that White's pattern inventory polynomials exhibit the cyclic sieving behavior known for the classical Pólya–Redfield polynomials in precisely the cases suggested by the cyclicity of the relevant unit group. Its status is unresolved in the supplied source.