Smooth representative conjecture for finite-energy homotopy sectors

Let $M$ be the source manifold, $G$ a Lie group, and $X=G/H$ a homogeneous space. Write $W_E^{1,2}(M,X)$ for the finite-energy maps and

$$\widetilde{W}_E^{1,2}(M,X):=\bigcup_{\varphi\in C^\infty}W_E^{1,2}(M,G)\varphi.$$

Smooth representative conjecture. Every $2$-homotopy sector of finite-energy maps contains a smooth representative; equivalently,

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

This would identify all finite-energy maps with the sectors represented through smooth reference maps and is one of the density and admissibility issues needed for a regularity theory; the source states it as unresolved in general.