Combining results of Gromov, McDuff, Taubes shows that any two symplectic forms on $\mathbb{C}P^2$ with the same total volume are symplectomorphic (e.g. to the Fubini-Study form $\omega_{FS}$). See for example McDuff-Salamon's book. Is there an analagous result for symplectic forms on $\mathbb{C}P^3$? McDuff-Salamon do not mention anything about this.
Suppose $\omega_1,\omega_2$ are symplectic forms on $\mathbb{C}P^3$ such that the almost complex structures $J_1,J_2$ tamed by these forms have $c_1(T\mathbb{C}P^3,J_1)=c_1(T\mathbb{C}P^3,J_2)$. Are the forms homotopic through symplectic forms? Symplectomorphic?
Any references would be appreciated, also for other symplectic 6-manifolds.