Boundary preservation under pointed limits of non-collapsed RCD spaces

Let $K\in\mathbb{R}$, $N\in\mathbb{N}$, and $v>0$. Let $\{(X_i,\mathsf d_i,\mathcal{H}^{N}_{\mathsf d_i})\}_{i\in\mathbb{N}}$ be a sequence of non-collapsed $\mathsf{RCD}(K,N)$ spaces such that

$$\{(X_i,\mathsf d_i,p_i)\}_{i\in\mathbb{N}}\longrightarrow (X,\mathsf d,p)$$

in pointed Gromov--Hausdorff sense, $\mathcal{H}^{N}(B_1(p_i))\geq v$ for every $i$, and $\boldsymbol\partial X_i=\emptyset$ for every $i$. Boundary-limit conjecture. The limit $(X,\mathsf d,\mathcal{H}^{N})$ is a non-collapsed $\mathsf{RCD}(K,N)$ space with $\boldsymbol\partial X=\emptyset$. The preceding theorem proves this under an additional quantitative-stratification hypothesis; the conjecture asserts that the hypothesis that every approximating space has empty boundary suffices.