The Hitchin-component conjecture for generalized complex-structure moduli

Let $\Sigma$ be the underlying surface, let $\mathfrak{g}$ be a real split Lie algebra, let $G$ be the real split Lie group associated to $\mathfrak{g}$, and let $\hat{\mathcal{T}}_{\mathfrak{g}}\Sigma$ be the moduli space of $\mathfrak{g}$-complex structures. Hitchin-component conjecture. The moduli space $\hat{\mathcal{T}}_{\mathfrak{g}}\Sigma$ is canonically homeomorphic to Hitchin's component in the character variety

$$\operatorname{Hom}(\pi_1(\Sigma),G)/G.$$

The conjecture identifies this generalized Teichmüller-type moduli space with the Hitchin component, whose shared dimension, contractibility, and Teichmüller copy motivate the proposal; it is not proved in the paper.