The minimal-solvable-orbit generalisation of Saxl's conjecture for Weyl groups

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$. Let $\mathcal{O}$ be the unique minimal solvable nilpotent orbit of $\mathfrak{g}$. For each Lusztig family $F$, let $\mathcal{O}_F$ be its attached special orbit. The minimal-solvable-orbit generalised Saxl conjecture. The representation

$$\left(\bigoplus_{\sigma\in F:\ \mathcal{O}\geq\mathcal{O}_F\geq d(\mathcal{O})}\sigma\right)^{\otimes 2}$$

contains every irreducible character of $W$.

This formulation uses the closure ordering on nilpotent orbits and is motivated by bounds on the Lusztig families that can occur in tensor products with spinor modules.