Conjecture on Springer fiber graph duality for special orbits in odd orthogonal and symplectic Lie algebras

Let $\mathbf{k}$ be the ground field, and let $N\in\operatorname{Lie}(SO_{2n+1})(\mathbf{k})$ and $N'\in\operatorname{Lie}(Sp_{2n})(\mathbf{k})$ lie in special nilpotent orbits related by the order-preserving bijection between the special nilpotent orbits of these two Lie algebras. Let $\Gamma_N$ denote the graph attached to the Springer fiber of $N$, and let the component group of the stabilizer of $N$ in $G$ act on $\Gamma_N$. The conjecture. The quotient of $\Gamma_N$ by the action of the component group of the stabilizer of $N$ in $G$ is isomorphic to the graph corresponding to $N'$. The question is motivated by the fact that corresponding orbits have the same irreducible Springer representation; no resolution is given in the source.