Alexandersson's minimal-polynomial conjecture for skew Schur functions

Let $\pmb{\kappa},\pmb{\lambda},\pmb{\mu},\pmb{\nu}$ be as in Theorem~. For every vector $\mathbf{w}$, let $\overline{\mathbf{w}}$ denote the vector obtained by rearranging its coordinates in non-increasing order, and define

$$W=\{\mathbf{w}\in\mathbb{N}^n\colon K_{\pmb{\lambda}/\pmb{\mu},\mathbf{w}}>0\text{ and }\overline{\mathbf{w}}\unrhd\overline{\pmb{\lambda}-\pmb{\mu}}\}.$$

For variables $\mathbf{x}$, write $\mathbf{x}^{\mathbf{w}}$ for the corresponding monomial. Alexandersson's conjecture. For sufficiently large $r$, the sequence

$$\{s_{\kappa+l\mu/\lambda+l\nu}(\mathbf{x})\}_{l=r}^{\infty}$$

satisfies a linear recursion with minimal polynomial

$$\chi(X)=\prod_{\mathbf{w}\in W}(X-\mathbf{x}^{\mathbf{w}}).$$

The conjecture concerns the eventual linear-recursive behavior of skew Schur functions and predicts the complete set of characteristic roots through the vectors in $W$. Its resolution is not established by the supplied text.