Jack fundamental quasisymmetric expansion conjecture

Let $n$ be a positive integer, let $\mu$ be a partition of size $n$, let $S_n$ be the symmetric group, and let $Q_A(x)$ denote the fundamental quasisymmetric function indexed by a set $A\subseteq\{1,\ldots,n-1\}$. Let $des(\pi)$ denote the number of descents of $\pi\in S_n$. Jack quasisymmetric expansion conjecture. There exists a set-valued function $\sigma$ depending on $\pi$, $\tau$, and $\mu$, with image in $\{1,\ldots,n-1\}$, such that

$$\tilde{J}_\mu^{(\alpha)}(X)=\sum_{\pi,\tau\in S_n}{\alpha+n-1-des(\pi)\choose n}Q_{\sigma(\pi,\tau,\mu)}(x).$$

The conjecture generalizes the known expansions for $\mu=(n)$ and $\mu=(1^n)$; the source also explains that, assuming Schur positivity in this basis, existence of some such $\sigma$ follows, but an explicit construction remains unavailable in general.