Bott ellipticity conjecture for positively curved manifolds

Let $M^n$ be a simply connected positively curved manifold, and let $\Lambda M$ denote its loop space. The Betti numbers of $\Lambda M$ are the homology ranks with coefficients in any chosen field. Ellipticity conjecture. The Betti numbers of the loop space of any simply connected $M^n \in \mathcal{P}_{0^{-}}$ grow at most polynomially. This is presented as an extremely strong size and structure restriction, as a proposed extension of a conjecture attributed to Bott. The source does not state a resolution, so the conjecture remains open.