The classical splitting theorem (Toponogov, Milka) says that if a smooth complete Riemannian manifold (more generally, Alexandrov space) $M^n$ of non-negative sectional curvature contains a line (i.e. infinite geodesic line minimizing distance between any pair of its points) then $M$ is isometric to $\mathbb{R}\times N^{n-1}$ where $N$ is non-negatively curved.

My question is whether there exists something in the spirit of the following quantitative version of the above theorem.

Let us fix $n\in \mathbb{N}$,$l>0$, $\delta>0$, and $r>1000$. Does there exist $\varepsilon>0$ and $L>0$ with the following properties. Let $M^n$ be a Riemannian manifold or Alexandrov space (not necessarily complete!) of curvature $\geq -\varepsilon$ such that it contains a compact ball $\bar B(x,r)$. Assume there exists a minimal geodesic of length $L$ such that $x$ is its middle point, and the whole space $M$ is contained in 1-neighborhood of this geodesic. Then there exists an open subset of the ball $\bar B(x,r)$ which is bi-Lipschitz homeomorphic to $(0,l)\times N^{n-1}$ where $N$ has curvature $\geq -\delta$.