Higher-rank harmonic map existence conjecture for framed surface-group representations

Let $X \in \mathcal{T}^{\pm}(\mathbb{S}, \mathbb{M})$ and let $\hat{X}$ be as before. Write $X_n={\mathrm{PSL}_n(\mathbb{C})}/{\text{PSU}(n)}$ for the symmetric space and let $\mathcal{F}(\mathbb{C}^n)$ denote the space of complete flags in $\mathbb{C}^n$. A framed representation $\hat{\rho}=(\rho,\beta): (\pi_1(\hat{X}),F_\infty)\to({\mathrm{PSL}_n(\mathbb{C})},\mathcal{F}(\mathbb{C}^n))$ is non-degenerate and type-preserving as in the source. Higher-rank harmonic map conjecture. Given such a non-degenerate type-preserving framed representation, there exists a $\rho$-equivariant harmonic map

$$h:\mathbb{H}^2\to X_n$$

that is asymptotic to $\beta$. Moreover, the map is unique if the principal parts of the Hopf differential at the punctures are prescribed. This is proposed as an analogue of the paper's existence and uniqueness theorem for framed ${\mathrm{PSL}_2(\mathbb{C})}$-representations, replacing $\mathbb{H}^3$ by the symmetric space $X_n$. Its status is not established in the supplied text.