Topological invariance of rationality for threefold Mori fibre spaces

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, with total spaces $X$ and $Y$ diffeomorphic. Topological rationality conjecture. If $X$ is rational, then $Y$ is also rational. This conjecture asks whether rationality of the total space is preserved by diffeomorphism among threefold Mori fibre spaces.