Humphreys' conjecture on Springer fiber graph duality for special nilpotent orbits

Let $G$ be a simple algebraic group over an algebraically closed field, let $\mathfrak{g}=\operatorname{Lie}G$, and let $N\in\mathfrak{g}$ lie in a special nilpotent orbit. 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 Lusztig–Spaltenstein dual of the orbit of $N$ is another special nilpotent orbit in $\mathfrak{g}$. Humphreys' 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 the Lusztig–Spaltenstein dual of the orbit. This extends the observed type-$A$ graph symmetry for transpose-related hook, two-row, and two-column partitions; the paper verifies the conjecture for minimal and minimal special nilpotent orbits, while the general assertion remains open.