Carrell's rationality conjecture for varieties with isolated vector-field zeroes

Let $X$ be a smooth (or nonsingular) irreducible projective algebraic variety admitting a holomorphic vector field whose zero scheme is nonempty and consists of isolated zeroes. Carrell's rationality conjecture. The variety $X$ is rational. This is a fundamental rationality problem. It is proved for surfaces and for varieties of dimension at most five, as well as under several additional hypotheses, but remains open in general.