The cyclic sieving conjecture for decorated permutations and extended codes

For a positive integer $n$, let $C_n$ be the cyclic group, let $\widetilde{{\mathfrak S}}_{n,j-1}$ be the set of decorated permutations with the specified excedance statistic, and let $\widetilde a_{n,j}(q)$ be the associated polynomial.

Cyclic sieving conjecture. For any positive integer $n$ and $0\leq j\leq n$, the triple

$$\left(\widetilde{{\mathfrak S}}_{n,j-1},~C_n,~\widetilde a_{n,j}(q)\right)$$

exhibits the cyclic sieving phenomenon.

The conjecture is motivated by the anticipated $C_n$-equivariant bijection between decorated permutations and extended Stembridge codes. The supplied text gives no resolution evidence, so its status remains open.