Canonical isomorphism between geometric Hitchin space and Hitchin's component

Let $\Sigma$ be a closed surface of genus $g\geq 2$, and let $\hat{\mathcal{T}}^n_{\Sigma}$ denote geometric Hitchin space and $\mathcal{T}^n_{\Sigma}$ Hitchin's component. Canonical isomorphism conjecture. Geometric Hitchin space

$$\hat{\mathcal{T}}^n_{\Sigma}$$

is canonically isomorphic to Hitchin's component

$$\mathcal{T}^n_{\Sigma}.$$

Both spaces are contractible real manifolds of the same dimension, but the proposed canonical isomorphism would relate the geometric structures and properties available on the two sides, including the complex structure and forgetful maps of geometric Hitchin space with the symplectic structure of Hitchin's component. No such isomorphism is established in the supplied context.