Petrie's conjecture on Pontryagin classes of homotopy complex projective spaces

Let $M$ be a $2n$-dimensional compact oriented manifold homotopy equivalent to the complex projective space $4\mathbb{CP}^n4$. Suppose that $M$ admits a non-trivial $S^1$-action. Petrie's conjecture. The total Pontryagin class of $M$ agrees with that of $\mathbb{CP}^n$. The conjecture is known in several special cases, including torus actions and dimensions up to at least $8$, but remains open in full generality.