Toric Specht module decomposition conjecture

Let $\kappa=\lambda/d/\mu$ be a toric shape, let $S^\kappa$ be its toric Specht module, and let $S^\nu$ denote the irreducible representation of the symmetric group indexed by $\nu\in P_{kn}$. Let $C_{\mu\nu}^{\lambda,d}$ be the corresponding Gromov–Witten invariants. Toric Specht module conjecture. The coefficients of the irreducible components in $S^{\lambda/d/\mu}$ are the Gromov–Witten invariants, namely

$$S^{\lambda/d/\mu}=\bigoplus_{\nu\in P_{kn}} C_{\mu\nu}^{\lambda,d}\,S^\nu.$$

This proposes a representation-theoretic, subtraction-free interpretation of the Gromov–Witten invariants and would give a decomposition rule for toric Specht modules analogous to the usual irreducible Specht-module theory. The source presents it as an open conjecture.