Negative curvature conjecture for minimal immersions from Hitchin representations

Let $\Sigma$ be the underlying surface and consider a Hitchin representation parametrized by $(0,q_3,\cdots,q_n)$. Let $f$ be the associated minimal immersion into the symmetric space $G/K$, and let $\sigma$ be the tangent space of the image of $f$. The sectional curvature of the immersed minimal surface is the curvature induced by the ambient symmetric-space sectional curvature $K_{G/K}(\sigma)$.

Negative curvature conjecture. The minimal immersion is never tangential to any flat inside the symmetric space. Consequently, the sectional curvature of the immersed minimal surface is strictly negative.

This extends the corresponding result established in the special $q_n$ and $q_{n-1}$ cases, where the Hitchin equation does not decouple and the tangent plane has strictly negative ambient sectional curvature. Its resolution is not specified in the source.