Admissible representative conjecture for finite-energy lifts

Let $M$ be a manifold, $G$ a Lie group, $X=G/H$, and let $\varphi$ be smooth. An admissible lift is a map $u$ whose gauge potential $u^{-1}du$ satisfies the admissibility conditions defining $\mathcal E(M,G)$. Let $W_E^{1,2}(M,G)$ denote the finite-energy lifts and let $\widetilde{W}_E^{1,2}(M,X)$ be the union of their images acting on smooth reference maps. Admissible representative conjecture. For any smooth $\varphi$ and finite-energy $u$, there is an admissible $\widetilde u\in\mathcal E(M,G)$ such that $u\varphi=\widetilde u\varphi$; equivalently,

$$\mathcal E(M,X)=\widetilde{W}_E^{1,2}(M,X).$$

This conjecture makes admissible maps coincide with finite-energy maps modulo the choice of lift; it is proved in the source when $H$ is a torus, but remains open for general homogeneous spaces.