The cycle and inverse descent-class fine-set conjecture

Let $n$ be a positive integer, let $J\subseteq [n-2]$, and let $D_{n-1,J}^{-1}\subseteq {\mathcal S}_n$ be the inverse descent class consisting of permutations that fix $n$. Let $C_n$ denote the $n$-cycle, and let ${\mathcal Q}$ denote the associated symmetric-function character. A multiset is fine when its associated character is represented by its quasisymmetric generating function as in the paper.

Cycle–inverse descent-class conjecture. For every $J\subseteq [n-2]$,

$${\mathcal Q}(C_n D_{n-1,J}^{-1})={\mathcal Q}(D_{n-1,J}^{-1}C_n).$$

Thus $C_nD_{n-1,J}^{-1}$ is a fine set for $S^{Z_{n-1,J}}\uparrow^{{\mathcal S}_n}$.

This is proposed as a far-reaching generalization of the equality for the relevant cycle and inverse descent-class products. The source notes that both products are sets, but that the analogous assertion fails when $C_n$ is replaced by a general fine set; the conjecture itself remains open.