Let $X$ be an $n$-dimensional Alexandrov space with curvature at least -1. Assume that at every point it has an $(n,\delta)$-strainer of length $\mu$, where $\delta$ and $\mu$ are independent of a point.

**Does there exist $\sigma=\sigma(\delta, \mu)$ such that for any geodesic triangle of diameter less than $\sigma$ if one of its angles is at least $\pi-1/1000$ then the other two are less than $1/10$?**

For smooth Riemannian manifolds with the injectivity radius at least $\mu$ the answer is positive as it was explained in the answer to this post: A property of geodesic triangles in manifolds with lower bounds on curvature and injectivity radius However I do not see how to generalize directly the argument.