Hubbard's surjectivity conjecture for the Hénon-map monodromy homomorphism

Let $\mathcal{H}^{\mathbb{C}}_0$ be the component of the complex Hénon hyperbolic locus containing $\mathrm{HOV}$, let $(a_0,c_0)\in\mathrm{DN}$ be a basepoint, and let $h_0:K_{a_0,c_0}^{\mathbb{C}}\to\Sigma_2$ be a topological conjugacy. For a loop $\gamma$ in $\mathcal{H}^{\mathbb{C}}_0$ based at $(a_0,c_0)$, let $\rho(\gamma)=h_1\circ(h_0)^{-1}$ be the associated monodromy homomorphism, and write $\Gamma$ for its image. Hubbard's conjecture. The monodromy homomorphism $\rho$ is surjective, that is, $\Gamma=\operatorname{Aut}(\Sigma_2)$. Surjectivity is known for the analogous polynomial-map monodromy, while the source presents the complex Hénon-map case as Hubbard's conjecture and gives no resolution.