The flat normal bundle conjecture for Ricci-limit spaces

Let $X$ be a Gromov–Hausdorff limit of complete noncompact Riemannian manifolds $M_i$ with $\operatorname{Ricci}(M_i)\geq -\epsilon_i$ and $\epsilon_i\to 0$. Such an $X$ is a complete, locally compact length space; the loops to infinity property means that every relevant loop can be homotoped arbitrarily far from a fixed basepoint. A split double cover is a double cover with the splitting property described in the preceding results, and a flat normal bundle over a compact totally geodesic soul is the geometric structure asserted below.

The flat normal bundle conjecture. If $X$ is a limit space as above, then either $X$ has the loops to infinity property or it has a split double cover and it is the flat normal bundle over a compact totally geodesic soul.

This conjecture would extend the previously known theorem for spaces without the loops to infinity property, whose proof relied on differentiation, to Ricci-limit spaces using recent differentiability results for metric measure spaces. Its status is not resolved in the supplied source context.