Small collapsed RCD spaces of nearly maximal rank are infranilmanifolds

For $K\rrbracketin\mathbb{R}$ and $N\in\mathbb{N}$, let an $\operatorname{RCD}(K,N+\varepsilon)$ space mean a metric-measure space satisfying the RCD condition with parameters $(K,N+\varepsilon)$, and let $\operatorname{rank}(\pi_1(X))$ denote the rank of its fundamental group. Nearly maximal-rank infranilmanifold conjecture. For each $K\in\mathbb{R}$ and $N\in\mathbb{N}$, there is $\varepsilon>0$ such that if $(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(K,N+\varepsilon)$ space with

$$\operatorname{diam}(X)\leq\varepsilon$$

and $\operatorname{rank}(\pi_1(X))=N$, then $X$ is homeomorphic to an infranilmanifold. The conjecture is motivated by examples showing that rigidity at the full rectifiable dimension can fail, while the source proposes that a sufficiently small dimension gap restores topological rigidity.