The universal-cover conjecture for extremally Ricci-pinched closed $G_2$ structures

Let $M$ be a compact manifold with a closed $G_2$ structure $\phi$ and extremally pinched Ricci curvature. Write $\widetilde{M}$ for the universal covering space of $M$, and let $G$ be the Lie group consisting of affine transformations of two-dimensional complex space preserving its standard complex volume form. The homogeneous space $G/\operatorname{\textsl{SU}}(2)$ carries a unique $G_2$ structure with parallel torsion.

Universal-cover conjecture. The universal covering space $\widetilde{M}$ is isometric to $G/\operatorname{\textsl{SU}}(2)$ equipped with its unique $G_2$ structure with parallel torsion.

This predicts that the extremally Ricci-pinched case is globally determined on the universal cover by the homogeneous model with parallel intrinsic torsion. The supplied text gives preceding classification and curvature-equality results but does not state that this conjecture has been resolved.