Existence of equivariant indecomposable minimal surfaces for negatively curved manifolds

Let $M$ be a closed oriented negatively curved Riemannian $n$-manifold with fundamental group $\Gamma$. An orthogonal representation $\rho:\Gamma\to \operatorname{End}(H)$ is weakly equivalent to the regular representation $\lambda_\Gamma$ when it has the same representation-theoretic equivalence class as $\lambda_\Gamma$. A minimal surface is indecomposable when it cannot be decomposed into smaller minimal surfaces in the relevant sense.

Existence conjecture. There exist an orthogonal representation $\rho:\Gamma\to \operatorname{End}(H)$ weakly equivalent to the regular representation $\lambda_\Gamma$, and a $\Gamma$-equivariant, indecomposable, $n$-dimensional minimal surface $\Sigma$ in the unit sphere of $H$.

This conjecture proposes that the spherical quotient appearing in the general Plateau theorem can be replaced by a smooth spherical quotient, paralleling the situation for negatively curved locally symmetric manifolds. The supplied text does not state whether the conjecture is known or open.