The spinorial generalised Saxl conjecture for Weyl groups

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$, reflection representation $V$, and Spaltenstein map $d$ on nilpotent orbits. Let $\Psi$ be the map from genuine irreducible representations of the Pin cover of $W$ to solvable nilpotent orbits. Suppose $\mathcal{O}$ is a solvable nilpotent orbit with $d(\mathcal{O})=\mathcal{O}$. The spinorial generalised Saxl conjecture. The representation

$$\bigwedge\nolimits^{\bullet}(V)\otimes\left(\bigoplus_{\widetilde{\sigma}\in\Psi^{-1}(\mathcal{O})}\widetilde{\sigma}\right)^{\otimes 2}$$

contains every irreducible character of $W$.

This is presented as an equivalent spin-representation formulation of the Weyl-group generalisation, arising from the interaction between genuine representations of the Pin cover, the spinor module, and the Springer correspondence.