Cayley correspondence conjecture for higher Teichmüller components

Let $X$ be a compact Riemann surface, let $\Sigma$ be its underlying surface, and let $\mathrm{G}^\mathbb{R}$ be a simple Lie group. Write ${\mathcal M}(X,\mathrm{G}^\mathbb{R})$ for the moduli space of $\mathrm{G}^\mathbb{R}$-Higgs bundles and ${\mathcal X}(\Sigma,\mathrm{G}^\mathbb{R})$ for the character variety. Under the nonabelian Hodge correspondence between these spaces, the Cayley components in ${\mathcal M}(X,\mathrm{G}^\mathbb{R})$ correspond to the higher Teichmüller components in ${\mathcal X}(\Sigma,\mathrm{G}^\mathbb{R})$. Cayley correspondence conjecture. Moreover, the list of Cayley components completes the classification of connected components in ${\mathcal M}(X,\mathrm{G}^\mathbb{R})$ for every simple Lie group $\mathrm{G}^\mathbb{R}$. This would identify all components arising from the Cayley correspondence with the higher Teichmüller components and settle whether the components found in the paper exhaust the components of $\Theta$-positive representations. The source presents this as an open conjecture.