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If true, such a statement would immediately imply the Poincare conjecture. Indeed, suppose X were a closed, simply connected 3-manifold with a Mobius structure. Near any point of $X$, there would then be a Mobius map from $X$ to $S^3$. Analytically continuing this map (see Thurston's notes, chapter 3) would give a local homeomorphism $\phi: X \to S^3$. Since both spaces are compact, $\phi$ would be a covering map. Since $S^3$ is simply connected it would be a homeomorphism. A similar argument applies to projective structures, with $S^3$ replaced by $\mathbb{RP}^3$.