The globalization conjecture for Alexandrov spaces

Let $Y$ be an $n$-dimensional length metric space. For a subset $\mathcal{U}\subseteq Y$, set $\mathcal{A}=Y\setminus\mathcal{U}$, and write $\mathcal{U}\in\operatorname{Alex}_{\mathrm{loc}}^n(\kappa)$ when every point of $\mathcal{U}$ has a neighborhood on which $\kappa$-Toponogov comparison holds. Globalization conjecture. If

$$\mathcal{H}^{n-1}(\mathcal{A})=0$$

and, for every $p\in\mathcal{A}$, the tangent cone $T_p(Y)$ exists and is isometric to a metric cone $C(\Sigma)\in\operatorname{Alex}^{n}(0)$, then $Y\in\operatorname{Alex}^{n}(\kappa)$. This would extend the stated globalization theorem, which assumes that the exceptional set is discrete, and would imply the paper's earlier gluing conjecture together with the results established in its separation and involutional gluing sections.