The Kostka-support conjecture for minimal recurrences of stretched skew Schur polynomials

Let ${\boldsymbol\kappa},{\boldsymbol\lambda},{\boldsymbol\mu},{\boldsymbol\nu}$ be partitions of length at most $n$ such that ${\boldsymbol\mu}\supseteq{\boldsymbol\nu}$ and $k({\boldsymbol\mu}-{\boldsymbol\nu})\supseteq{\boldsymbol\kappa}-\boldsymbol\lambda$ for some positive integer $k$. For a vector $\mathbf{w}\in\mathbb{N}^n$, let $K_{{\boldsymbol\mu}/{\boldsymbol\nu},\mathbf{w}}$ denote the corresponding Kostka coefficient, and let $\overline{\mathbf{w}}$ be the rearrangement of $\mathbf{w}$ into a partition; write $\trianglerighteq$ for dominance order. Define

$$W=\{\mathbf{w}\in\mathbb{N}^n\mid K_{{\boldsymbol\mu}/{\boldsymbol\nu},\mathbf{w}}>0\ \wedge\ \overline{\mathbf{w}}\trianglerighteq\overline{{\boldsymbol\mu}-{\boldsymbol\nu}}\}.$$

The Kostka-support conjecture. For sufficiently large $r$, the sequence

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

satisfies a linear recurrence with minimal characteristic polynomial

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

This is proposed as a sharper version of the paper's greedy recurrence theorem, replacing its potentially larger root set by the support determined by Kostka coefficients and dominance order; its resolution is not supplied here.