Let $(M,g)$ be a Riemannian manifold and suppose that the Weyl tensor of $g$ vanishes at a point $p \in M$. Can one estimate the size of the largest geodesic ball around $p$ that we can make $g$ flat on through conformal deformation in terms of geometric data, i.e., $g$ and the usual curvature invariants? Is this just a function of the injectivity radius?

Also: suppose $g$ is locally conformally Einstein. Can we produce similiar estimates on the largest geodesic ball around a given point that we can make $g$ Einstein on?