The higher complex structures conjecture for the Hitchin component

Let $\Sigma$ be a closed oriented surface, let $\mathcal{T}^n$ be the moduli space of higher complex structures on $\Sigma$, and let

$$\omega:T^*\mathcal{T}^n\to \operatorname{Rep}(\pi_1\Sigma,\operatorname{SL}_n(\mathbb{C}))$$

be the formal map whose image is an open subset of the character variety. The higher complex structures conjecture. The restriction of $\omega$ to the zero-section $\mathcal{T}^n\subset T^*\mathcal{T}^n$ has its image inside the real character variety

$$\operatorname{Rep}(\pi_1\Sigma,\operatorname{SL}_n(\mathbb{R})).$$

Hence, there is a canonical homeomorphism between $\mathcal{T}^n$ and the Hitchin component. This would provide a geometric approach to higher Teichmüller theory; the paper does not establish the conjecture, and the convergence of the reconstruction is not analyzed.