The negative curvature conjecture for Hitchin and maximal representations

Let $\rho$ be a Hitchin representation into $\operatorname{PSL}(n,\mathbb R)$ or a maximal representation into $\operatorname{Sp}(2n,\mathbb R)$, let $f$ be the associated equivariant harmonic map, and at an immersed point let $\sigma$ be the tangent plane to its image, $k_\sigma^N$ the sectional curvature of the target symmetric space along $\sigma$, and $\kappa$ the Gaussian curvature of the induced metric.

Negative curvature conjecture. The sectional curvature $k_\sigma^N$ is negative at every immersed point. Consequently, the induced curvature $\kappa$ is negative at every immersed point.

Geometrically, this says that the harmonic-map image is never tangent to a flat in the symmetric space. The source identifies the $\operatorname{PSL}(n,\mathbb R)$ case for minimal maps with the negative curvature conjecture of Dai and Li; the general statement remains open in the source.