Lehrer–Shoji's exterior-power conjecture for Springer fibres

Let $\mathfrak{g}$ be a simple complex Lie algebra of rank $\ell$, let $W$ be its Weyl group, and let $V$ be the reflection representation of $W$. Let $e$ be a nilpotent element, and let $\mathcal{B}_e$ be the Springer fibre of Borel subalgebras containing $e$. Write $s$ for the multiplicity of $V$ in $H^*(\mathcal{B}_e)$, and let $m_1,\ldots,m_s$ be the halved degrees of its occurrences, so that

$$\sum_j \dim\bigl(H^{2j}(\mathcal{B}_e)\otimes V\bigr)^W q^j=q^{m_1}+\cdots+q^{m_s}.$$

Lehrer–Shoji's conjecture. Suppose that $e$ is a regular nilpotent in a Levi subalgebra. Then, for every $i=0,1,\ldots,\ell$,

$$\sum_j \dim\bigl(H^{2j}(\mathcal{B}_e)\otimes \Lambda^i V\bigr)^W q^j=e_i(q^{m_1},q^{m_2},\ldots,q^{m_s}),$$

where $e_i$ is the $i$th elementary symmetric polynomial, defined to be zero if $i>s$. The conjecture predicts that the graded occurrences of all exterior powers of the reflection representation are determined by the degrees $m_1,\ldots,m_s$; its precise status is not established by the supplied text.