Let $X$ be a Hadamard manifold with curvature $-b^2 \le K \le -1$. Let $c$ be a complete geodesic in $X$. For every $x \in X$, denote by $p_c(x)$ the projection of $x$ onto $c$. This projection map being $1$-Lipschitz, one has $d(p_c(x), p_c(y)) \le d(x, y)$ for every $x, y$.
Here's my question : if we know that $d(x, p_c(x)) \le r$ and $d(y, p_c(y)) \le r$, can we give an estimate about how fast $d(x, y) - d(p_c(x), p_c(y))$ goes to $0$ when $r$ goes to $0$ ? The triangle inequality yields $$ d(x, y) - d(p_c(x), p_c(y)) \le 2 r$$ but I'd like to at least have a $(1+\varepsilon)$ instead of this $2$, possibly after assuming that $d(x, y)$ is large enough.