Quantified asymptotic Euclideanity conjecture for RCD spaces

Let $(X,d,m)$ be an $\mathrm{RCD}(0,N)$ space, let $p\in X$, and let $h_{1+\delta}(X,p)=k$ with $k\leq N$. For $\epsilon>0$ and $N\geq 1$, consider the parameter $\delta(N,\epsilon)>0$.

Quantified asymptotic Euclideanity conjecture. There exists $\delta(N,\epsilon)>0$ such that, for every $\delta\in(0,\delta(N,\epsilon)]$, there exists $R_0\geq 1$ for which $B_R(p)$ is $(\epsilon,k)$-Euclidean for every $R\geq R_0$.

This is presented as a quantified version of the splitting-at-infinity result and extends the preceding proposition from complete manifolds with non-negative sectional curvature to non-negatively curved Alexandrov and, conjecturally, $\mathrm{RCD}$ spaces. The source does not provide a resolution of the conjecture.