Let us assume that this is not the case and study such a limit space $\mathds{R}^{n-3}\times C(Y)$. The first observation is that by Corollary 5.2, it follows that $Y$ is a smooth surface. Indeed, if there were a point $y\in Y$ such that $r_{h}(y)=0$. Then since $X=\mathds{R}^{n-3}\times C(Y)$ it would follow that there is a set of codimension at least $2$ such that $r_{h}\equiv 0$, which cannot happen by Corollary 5.2.

Since $Y$ a $C^{1,\alpha}\cap W^{2,q}$ manifold and $|{\rm Ric}_{M^{n}_{j}}|\to 0$, it follows that $Y$ is a smooth Einstein manifold satisfying ${\rm Ric}_{Y}=g$. Because $Y$ is a surface, this means in particular that $Y$ has constant sectional curvature $\equiv 1$. Thus, either $Y=\mathds{R}\mathds{P}^{2}$ or $Y=S^{2}$, the unit $2$-sphere, and in the latter case we are done.

So let us study the case $Y=\mathds{R}\mathds{P}^{2}$. For $\epsilon>0$ small, choose $u_{j}:B_{2}(p_{j})\to\mathds{R}^{n-3}$ to be an $\epsilon$-splitting as in Lemma 1.7. Note that away from the singular set $\mathcal{S}\equiv\mathds{R}^{n-3}\times\{0\}$ we have that the $M_{j}^{n}$ converge to $\mathds{R}^{n-3}\times C(Y)$ in $C^{1,\alpha}$. If $f_{j}:B_{2}(p)\to B_{2}(p_{j})$ denote the Gromov-Hausdorff maps, we put $\mathcal{S}_{j}=f_{j}(\mathcal{S})$. Then for $\tau>0$ small but fixed, we have for $j$ sufficiently large, that on $B_{1}(p)\setminus T_{\tau}(\mathcal{S}_{j})$, the estimates $|\nabla u_{j}|>\frac{1}{2}$ and $|\nabla^{2}u_{j}|\leq 1$ hold.

Consider Poisson approximation $h_{j}$ to the square of distance function $d^{2}(x,p_{j})$ on $B_{2}(p_{j})$. That is, $\Delta h_{j}=2n$ and $h_{j}=1$ on $\partial B_{2}(p_{j})$. We have (see for instance [ChCo1]) that $|h_{j}-d(\cdot,p_{j})|\to 0$ uniformly in $B_{2}(p_{j})$, and again because the convergence is in $C^{1,\alpha}$ we have for $j$ sufficiently large that $|\nabla h|>\delta$ and $|\nabla^{2}h|\leq 4n$ on $B_{1}(p)\setminus B_{\tau}(\mathcal{S}_{j})$. Once again, appealing to the $C^{1,\alpha}$ convergence, for all $j$ sufficiently large and all $s\in B_{1}(0^{n-3})$ we have that $u^{-1}(s)\cap h^{-1}(1)$ is diffeomorphic to $\mathds{R}\mathds{P}^{2}$. By Sard’s theorem, there exists a regular value $s_{j}\in B_{1}(0^{n-3})$. Then for $j$ sufficiently large, $u^{-1}_{j}(s_{j})\cap\{h\leq 1\}$ is a smooth $3$-manifold, whose boundary is diffeomorphic to $\mathds{R}\mathds{P}^{2}$. However, the second Stiefel-Whitney number of $\mathds{R}\mathds{P}^{2}$ is nonzero, and in particular, $\mathds{R}\mathds{P}^{2}$ does not bound a smooth $3$-manifold. This contradicts $Y=\mathds{R}\mathds{P}^{2}$.

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