The generalized Kostka–Foulkes and ribbon-tableau polynomial conjecture

Let $R=(R_1,\ldots,R_p)$ be a dominant sequence of rectangular partitions, and let $\Lambda$ be the partition with empty $p$-core and $p$-quotient $(R_1,\ldots,R_p)$. The polynomial $\widetilde{K}_{\Lambda\lambda}^{(p)}(q)$ is defined by the Schur expansion of the modified $p$-ribbon Hall–Littlewood function. Generalized Kostka–Foulkes conjecture.

$$\widetilde{K}_{\lambda;R}(q)=\widetilde{K}_{\Lambda\lambda}^{(p)}(q).$$

This identifies the generalized Kostka polynomials with the $q$-analogue arising from spin-generating functions of ribbon tableaux; the source presents it as a conjectural connection between two constructions.