The Alexandrov gluing conjecture

Let $Y$ be obtained by gluing boundaries of $n$-dimensional Alexandrov spaces. For each $p\in Y$, define the tangent cone, when it exists, by

$$T_p(Y)=\lim_{r\to 0}(Y,p,r^{-1}d).$$

A metric cone over a space $\Sigma_p$ is required to have $\Sigma_p\in\operatorname{Alex}^{n-1}(1)$.

Alexandrov gluing conjecture. The gluing produces an Alexandrov space $Y$ if and only if the gluing is by isometry and, for every $p\in Y$, $T_p(Y)$ is a metric cone over $\Sigma_p$.

This conjecture proposes a local tangent-cone characterization of when gluing Alexandrov spaces along their boundaries preserves the Alexandrov condition. It is motivated by the limitations of existing gluing theorems for more than two spaces or for gluing loci that are not the full boundary; the source gives no resolution.