The generic parameterization conjecture for flat parabolic connections

Let $\Sigma$ be a surface, and let $\mathcal{A}(\lambda)$ be a family of $\operatorname{PSL}_n(\mathbb{C})$-connections satisfying

$$d\mathcal{A}(\lambda)+\mathcal{A}(\lambda)\wedge\mathcal{A}(\lambda)=0,$$

with $\mathcal{A}(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^*$, principal-nilpotent $(1,0)$-part $\Phi_1$, and reality condition $-\mathcal{A}(-1/\bar\lambda)^*=\mathcal{A}(\lambda)$. Generic parameterization conjecture. There is an open dense subset of such connections which, modulo a choice of unitary gauge and a higher complex structure on $\Sigma$, is parameterized by holomorphic differentials $t_k\in H^0(K^k)$ for $k=2,\ldots,n$ and some finite data. This would give a higher-complex-structure formulation of the generic part of the relevant flat-connection moduli space; no resolution is stated.