Bounded conic intersection conjecture for sufficiently Euclidean Riemannian metrics

Let $\Phi(x,y)=\rho(x,y)$, where $\rho$ is a Riemannian metric sufficiently close to the Euclidean metric. The bounded conic intersection conjecture. Then $\Phi$ satisfies the bounded conic intersection property. The bounded conic intersection property is an analogue of the fact that distinct irreducible Euclidean conic sections intersect in at most $O(1)$ points. General defining functions satisfying the cinematic curvature hypotheses need not have this property, while the conjecture asserts it for metrics sufficiently close to Euclidean.