Openness and closedness conjecture for Θ-positive Anosov representations

Let $\Sigma$ be a closed oriented surface and let $(\mathrm{G},\mathrm{P}^\mathbb{R}_\Theta)$ admit a $\Theta$-positive structure. A representation $\rho:\pi_1\Sigma\to\mathrm{G}^\mathbb{R}$ is $\Theta$-positive Anosov when it is $\mathrm{P}^\mathbb{R}_\Theta$-Anosov and its Anosov boundary curve sends positively ordered triples in $\partial_\infty\pi_1\Sigma$ to positive triples in $\mathrm{G}^\mathbb{R}/\mathrm{P}^\mathbb{R}_\Theta$. Openness and closedness conjecture. The set of $\Theta$-positive Anosov representations is an open and closed subset of ${\mathcal X}(\Sigma,\mathrm{G}^\mathbb{R})$. This asserts that positivity defines a union of connected components of the character variety, a foundational property for the higher Teichmüller spaces considered in the paper. The source provides no resolution evidence, so the conjecture is recorded as open.