The factorization conjecture for Jack Littlewood–Richardson coefficients

Let $R$ be a window and let $\kappa$ be a factorization region with $\kappa\subseteq\mu_R\subset\lambda_R$, such that $\lambda_R/\kappa$ has multiple components $\{\lambda_i\}$ and $\mu_R/\kappa$ has components $\{\mu_j\}$. Let $\mathcal H_\kappa=\mathcal H_{\kappa,\emptyset}^{\kappa}$ be the set of factorization channels, let $G_{\Phi,R}^{\kappa}$ be the evaluated channel factor, and let $H_\Phi(\vec v)$ be a channel solution depending on local parameters $\vec v$.

Factorization conjecture. The Jack Littlewood–Richardson coefficient has an expansion

$$g_{\mu\nu}^{\lambda}=F_R\sum_{\Phi\in\mathcal H_\kappa}G_{\Phi,R}^{\kappa}H_\Phi(\vec v).$$

The channel set is independent of $R$, while the channel solutions are local to the component shapes and $\nu_R$. This conjecture is presented as a window-compatible generalization of a basic factorization theorem and is not proved in the source.