The inverse descent-class commutation conjecture for fine sets

Let ${\mathcal B}\subset {\mathcal S}_n$ be a fine set, and let $J\subseteq [n-1]$. Write $D_{n,J}^{-1}$ for the inverse descent class indexed by $J$, 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.

Inverse descent-class commutation conjecture. For every fine set ${\mathcal B}\subset {\mathcal S}_n$ and every $J\subseteq [n-1]$,

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

In particular, the multiset $D_{n,J}^{-1}{\mathcal B}$ is fine.

The conjecture concerns multiset products of fine sets with inverse descent classes and is supported by computer experiments. The stated consequence follows from the paper's main result on products with inverse descent classes; the general equality remains open.