Non-realizability of the minimal nilpotent orbit closure of type E₆ as a Hamiltonian reduction

Let $\overline{\mathcal{O}}_\textrm{min}^{E_6}$ be the closure of the minimal nilpotent orbit of the complex simple Lie algebra of type $E_6$. Let $V$ be a representation of a reductive complex algebraic group $G$, and let the Hamiltonian reduction of $T^*V$ by $G$ be formed with respect to the canonical moment map.

Non-realizability conjecture. The closure $\overline{\mathcal{O}}_\textrm{min}^{E_6}$ is not isomorphic to any Hamiltonian reduction of the cotangent space $T^*V$ of a representation $V$ by a reductive complex algebraic group $G$.

This asserts that the minimal nilpotent orbit closure in type $E_6$ cannot arise from the standard Hamiltonian-reduction construction, distinguishing it from orbit closures that do admit such realizations. The source provides no resolution status for this claim.