Singular nilpotent fibration conjecture

Suppose a sequence $\{(M_i,g_i)\}\subset \widetilde{\mathcal{M}}_{\rm Rc}(m,\rho,v)$ collapses to a lower-dimensional compact metric space $X$, with

$$M_i^m\xrightarrow{\rm GH}X^k,$$

where $k$ is the dimension in the sense of Colding--Naber. A singular fibration is a fibration $f_i\colon M_i\to X$ whose regular fibers and singular fibers have the types described below. Singular nilpotent fibration conjecture. For every sufficiently large $i$, there is such a singular fibration $f_i\colon M_i\to X$ in which every regular fiber is an infranil manifold and every singular fiber is a finite quotient of an infranil manifold. When $k=m$, equality of the Colding--Naber and Hausdorff dimensions is known; for $k<m$, the corresponding dimension equality remains open, as does this proposed singular-fibration picture.