Nash's rationality conjecture for compact smooth manifolds

Let $M$ be a compact, connected, boundaryless smooth manifold. A real algebraic variety $X$ is rational if it is birational to projective space over the real numbers, and $X(\mathbb{R})$ denotes its real locus.

Nash's rationality conjecture. There exists a rational variety $X$ whose real locus is diffeomorphic to $M$:

$$M\approx X(\mathbb{R}).$$

This strengthens the Nash–Tognoli theorem, which realizes every compact boundaryless smooth manifold as the real locus of a nonsingular projective real algebraic variety. The supplied text does not state whether Nash's stronger rationality conjecture is resolved.