The asymptotic-cone orbit conjecture for nonnegative Ricci curvature

Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not=1/2$. Suppose that $\pi_1(M)$ contains a torsion-free nilpotent subgroup of nilpotency length $l$. An asymptotic-cone orbit conjecture. There exist an asymptotic cone $(Y,y)$ of the universal cover and a closed $\mathbb{R}$-subgroup of $\operatorname{Isom}(Y)$ such that the orbit $\mathbb{R}y$ is homeomorphic to $\mathbb{R}$ but has Hausdorff dimension at least $l$. The conjecture is motivated by the study of equivariant asymptotic cones and is intended to explain how nilpotent structure in the fundamental group can produce highly dimensional asymptotic orbits despite their being topologically one-dimensional; its resolution is not given here.