The hyperkähler twistor conjecture for higher complex structures

Let $\Sigma$ be a smooth surface, let $n$ be a positive integer, and let $\mathcal{T}^n$ be the moduli space of higher complex structures of order $n$ on $\Sigma$. Write $T^*\mathcal{T}^n$ for its cotangent bundle, and let $U$ be an open subset of $\operatorname{Rep}(\pi_1\Sigma,\operatorname{SL}_n(\mathbb{C}))$. Hyperkähler twistor conjecture. The space $T^*\mathcal{T}^n$ has a hyperkähler structure near the zero-section; all but two fibers in the twistor space are diffeomorphic to $U$, and the zero-section corresponds to the Hitchin component. This conjectural picture is intended as a non-abelian-Hodge-type description of representation spaces. The source gives no evidence that the claimed structure or identifications have been proved.