Guichard-Labourie-Wienhard conjecture on positive representations

Let $\Sigma_g$ be a closed surface of genus $g$, let $G$ admit a $\Theta$-positive structure, and let $P_\Theta<G$ be the associated parabolic subgroup. A representation $\rho:\pi_1(\Sigma_g)\to G$ is $\Theta$-positive if there exists a continuous $\rho$-equivariant map

$$\xi:S^1\to G/P_\Theta$$

which sends positive triples in $S^1$ to positive triples in $G/P_\Theta$. The ambient quotient is $\operatorname{Hom}(\pi_1(S),G)/G$.

Guichard-Labourie-Wienhard conjecture. The set of $\Theta$-positive representations $\rho:\pi_1(\Sigma_g)\to G$ is open and closed in $\operatorname{Hom}(\pi_1(S),G)/G$. In particular, $\Theta$-positive representations form higher Teichmüller spaces.

The conjecture would produce higher Teichmüller spaces for all simple Lie groups admitting a $\Theta$-positive structure. The source lists the relevant families of groups and does not state that the conjecture is resolved.