Cyclic c-equivalence conjecture for centrally symmetric convex sets

Let $\Gamma\subset[k-1]\times[n-k-1]$ be centrally symmetric and convex. For $f\in\Theta_{k,n}$, let $\Gamma'(f)$ denote the associated centrally symmetric convex set, and let c-equivalence be the equivalence relation on such $f$. Let $\sigma$ be the cyclic-shift map.

Cyclic c-equivalence conjecture. The set

$$\{f\in\Theta_{k,n}\mid \Gamma'(f)=\Gamma\}$$

is a union of $\gcd(k,n)$ c-equivalence classes. These classes are cyclically permuted by $\sigma$.

The conjecture is intended to show that the objects $C_f$, $\widetilde R_f(q)$, $\mathcal{P}(\Pi^\circ_f;q,t)$, and $\hat\beta_f$ depend only on $\Gamma'(f)$. The source reports partial results and experimental evidence, but no general resolution.