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Ian Agol
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As I'm sure you're aware, the existence of such a flip would give rise to a counterexample to the 4-dimensional smooth Poincare conjecture (a 4-manifold homeomorphic but not diffeomorphic to $S^4$). So I think it can be safely assumed that this is an open problem.

Ian Agol
  • 68.9k
  • 3
  • 194
  • 358