The generalised Saxl conjecture for Weyl groups

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$, Lie group $G$, nilpotent cone $\mathcal{N}$, and set of nilpotent orbits $G\backslash\mathcal{N}$. Let $d:G\backslash\mathcal{N}\to G\backslash\mathcal{N}$ be Spaltenstein's order-reversing map, involutive on its range. Let $\mathcal{O}$ be a solvable nilpotent orbit corresponding, via the Springer correspondence with the trivial local system, to a special character $\sigma_F=\sigma_{\mathcal{O},\operatorname{triv}}$ in Lusztig family $F$, and suppose $d(\mathcal{O})=\mathcal{O}$. The generalised Saxl conjecture for Weyl groups. The representation

$$\left(\bigoplus_{\sigma\in F}\sigma\right)^{\otimes 2}$$

contains every irreducible character of $W$.

This generalises Saxl's conjecture from symmetric groups to Weyl groups using special nilpotent orbits and Lusztig families; the source presents it as a conjecture motivated by the Springer correspondence and spin representations.