The regular factorization conjecture for local Schur channel coefficients

For a factorization region $\kappa$, suppose the channel solutions are regular, so that the limits $C_\Phi(\vec v)$ at $\alpha\to1$ exist and the Schur coefficient decomposes as $c_{\mu\nu}^{\lambda}=\sum_{\Phi\in\mathcal H_\kappa}C_\Phi(\vec v)$.

Regular factorization conjecture. For certain factorizations $\kappa$, there exist factorization solutions for which

$$C_\Phi(\vec v)\in\mathbb Z$$

and each $C_\Phi(\vec v)$ is independent of $\vec v$. This predicts integer, local-parameter-independent contributions to the Schur Littlewood–Richardson coefficient for suitable factorizations. The source does not specify which factorizations qualify or establish the claim generally.