Topological invariance of rigidity for threefold Mori fibre spaces

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, and let rigidity mean birational rigidity of the Mori fibre space. Topological rigidity conjecture. Whether a threefold Mori fibre space is rigid is a topological property. Thus rigidity should be preserved under diffeomorphism. The source presents this as an optimistic conjecture, with the related claim that rationality and rigidity should be constant in algebraic families whose fibres are all diffeomorphic.