The conjectural order of the unipotent centralizer quotient in finite classical groups

Let $G$ be either ${\rm Sp}(V)$ or $\Omega(V)$, let $u$ be the representative in

, and write $P=QL$ for the canonical parabolic subgroup containing $C_G(u)$, with unipotent radical $Q$. Let $R$ be the normal subgroup of $C_G(u)$ of order $q^{M_u}$. Define

J={j:b_j=2}.

For $j\in J$, let $a_j$ be the multiplicity of $W(2k_j)$ in

. The centralizer quotient order conjecture. One has

$$|RC_Q(u)/C_Q(u)|=q^t,\qquad t=\sum_{j\in J}(2a_j+1).$$

This conjecture specifies the order of the quotient measuring the contribution of the normal subgroup $R$ inside the centralizer of $u$ and the unipotent radical of the canonical parabolic. The supplied source gives no resolution status, so it remains open.