Quasisymmetric characteristic formula for a statistic-and-descent model

Let $M$ be an $\fS_n$-module, and let $X$ be a set of combinatorial objects equipped with functions $\operatorname{stat}:X\to\mathbb{N}$ and $\operatorname{set}:X\to\mathcal{P}(\{1,2,\ldots,n-1\})$. Quasisymmetric characteristic formula conjecture. Then

$$F_{char}(M;q)=\sum_{x\in X}q^{\operatorname{stat}(x)}F_{\operatorname{set}(x)}.$$

This is presented as a general form of open conjectures motivated by deforming symmetric-group modules into $\mathcal{H}_n(0)$-modules; the excerpt does not specify additional hypotheses relating $M$ and $X$.