For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.

My question is as follows. Let $M$ be a closed symplectic manifold which is homotopy equivalent to a complex projective space. Can we say that $M$ is homeomorphic or diffeomorphic to the standard one?