Definition. A smooth map $f\colon M\to N$ of two smooth Riemannian manifolds is called $\varepsilon$-almost Riemannian submersion (here $0\leq \varepsilon<1/100$) if for any point $x\in M$ and any tangent vector $\xi\in T_xM$ which is orthogonal to the fiber $f^{-1}(f(x))$ one has $$(1-\varepsilon) ||df(\xi)||\leq ||\xi||\leq (1+\varepsilon)||df(\xi)||.$$

Question. Given an $\varepsilon$-Riemannian submersion as above. Does there exist $\delta >0$ with the following property? Given any point $x\in M$ and a geodesic $\gamma\colon [a,b]\to M$ whose endpoints belong to the fiber $f^{-1}(f(x))$. Assume that this geodesic is $\varepsilon$-almost orthogonal to the former fiber at its initial point $a$, i.e. the angle $\alpha$ between $\gamma'(a)$ and the tangent space to the fiber satisfies $|\alpha-\pi/2|<\varepsilon.$ Then the image $f(\gamma)$ of the geodesic cannot be contained in the $\delta$-neighborhood of the point $f(x)$.

For my applications $\delta$ might depend on $\varepsilon$, any geometry of $N$, and a lower bound on sectional curvature of $M$ (but not on other geometric properties of $M$).

Remark. For $\varepsilon=0$ the answer is known to be positive, and $\delta$ can be taken less then the injectivity radius of $N$.