Embedding conjecture for Hitchin harmonic maps

Let $G$ be a complex adjoint simple Lie group, let $\rho\in \operatorname{Hit}(G)$ be a Hitchin representation, and let $f_{\rho}:\widetilde{X}\rightarrow G/K$ be the corresponding $\rho$-equivariant harmonic map.

Embedding conjecture. The map $f_{\rho}$ is an embedding.

In rank one, this is the theorem of Schoen–Yau; the conjecture asks whether the analogous embedding property holds for Hitchin harmonic maps in higher rank.