The Alexandrov globalization conjecture

Let $(X,d_X)$ be a length metric space and let $\mathcal U\subset X$ be open, with

$$\mathcal H^{n-1}(X\setminus\mathcal U)=0.$$

Assume that $(\mathcal U,d_{\mathcal U})\in\operatorname{Alex}^n_{\mathrm{loc}}(\kappa)$. For every $p\in X$, suppose the tangent cone exists and is isometric to a metric cone $C(\Sigma_p)$:

$$T_pX=\lim_{r\to 0}(X,p,r^{-1}d),$$

where $\Sigma_p\in\operatorname{Alex}^{n-1}(1)$.

Alexandrov globalization conjecture. Then $(X,d_X)\in\operatorname{Alex}^n(\kappa)$.

This conjecture seeks to globalize a local Alexandrov lower-curvature bound under an almost-everywhere open-subset hypothesis and a prescribed tangent-cone structure at every point. It is inspired by the Globalization Problem and related conjectures; the source states that the claim is unresolved.