Abstract: A pair of points x,y in a Riemannian manifold (M,g) is said to be secure if there exists a finite set of points intercepting every geodesic segment joining x to y. Given any conformal equivalence class C of Riemannian metrics on a closed manifold M of dimension at least two and given any pair of points x, y in M, there exists a dense G_{δ} set C′ ⊂ C such that x and y are not secure for every metric g in C′.