The Gluing Conjecture for multiple Alexandrov spaces

Let $X\in\operatorname{Alex}_{\amalg}^n(\kappa)$ satisfy the paper's condition, let $Y$ be a connected length metric space, and let $f\colon X\to Y$ be a 1-Lipschitz onto map. Gluing Conjecture. The space $Y$ belongs to $\operatorname{Alex}^n(\kappa)$ if and only if the gluing induced by $f$ is by path isometry along the boundary and, for every $y\in Y$ and every limit gluing map $f_\infty$, the tangent cone $T_{f^{-1}(y)}(X)$ glues to an Alexandrov space. This gives necessary and sufficient conditions for a finite gluing of Alexandrov spaces to remain Alexandrov; the abstract formulation describes the same proposed criterion in terms of path-isometric boundary gluings and glued tangent cones.