Let $\Gamma\subset PSL(2,R)$ be a Fuchsian group. For which representations $\rho:\Gamma\to PSL(2,R)$ does there exist a harmonic map from the hyperbolic plane to itself satisfying
for all $\gamma\in \Gamma$?
I was told that when $\Gamma$ is co-compact and torsion free, the existence of such an $f$ was given by Simpson's correspondence. I am interested to know if this extends in any way to more general $\Gamma$, especially co-finte volume $\Gamma$, and if this fact can be deduced in an elementary way.