Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $\geqslant -1$. For a point $p\in M$, suppose there exist $l, \delta>0$, $x,y \in M$ with $d(p,x),d(p,y)>l$ and a geodesic $px$ and $py$ with angle $\angle xpy>\frac{\pi}{2}+\delta$. Let $q$ be a point on geodesic $px$ or $py$, Question: is there $r>0$, which depends only on $n,l,\delta$ such that $B_q(r)$ is homeomorphic to $B_p(r)$?

Equivalently, we can state the question in the following way:

Let $M_i$ be a sequence of Riemannian manifolds with $sec \geqslant -1$ and diameter $\leqslant D$. Suppose $(M_i,p_i)$ Gromov-Hausdorff converge (possibly collapse) to $(X,p)$ (we know it's an Alexandrov space). Suppose there exist $l>0, \delta>0$, $x,y\in X$ with $\angle xpy> \frac{pi}{2}+\delta$. lift $x,y$ to $M_i$, we get $x_i,y_i\in M_i$. with$\angle x_i p_i y_i >\frac{\pi}{2}+\delta$. Let $q_i$ be a point on geodesic $p_ix_i$ or $p_iy_i$. Question: Is there $r>0$, such that such that $B_{q_i}(r)$ is homeomorphic to $B_{p_i}(r)$?