Exterior-algebra conjecture for invariant Springer cohomology

Let $G$ be a simple connected adjoint algebraic group over an algebraically closed field of good characteristic, with Lie algebra $\mathfrak g$, Weyl group $W$, rank $n$, and irreducible reflection representation $V$. For a nilpotent element $e\in\mathfrak g$, let ${\mathcal B}_e$ be the Springer fiber and let $H^*({\mathcal B}_e)$ carry its Springer $W$-representation. Consider the $W$-invariant part of the bigraded algebra

$$\left(\bigoplus_{i=0}^n H^*({\mathcal B}_e)\otimes\wedge^iV\right)^W.$$

Invariant exterior-algebra conjecture. Except when $e=F_4(a_3)$ in type $F_4$ or $e=E_8(a_7)$ in type $E_8$, this algebra is an exterior algebra on the subspace

$$\left(H^*({\mathcal B}_e)\otimes V\right)^W.$$

This conjecture describes the exceptional simplicity of the $W$-invariants in the tensor product of Springer cohomology with the exterior algebra of the reflection representation. The paper proves the related Lehrer–Shoji conjecture in all types, but the supplied text does not state that this exterior-algebra conjecture itself is resolved.