In the paper 'On the complex projective spaces' of Hirzbruch and Kodaira,they prove that

If $X$ is compact Kahler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ in case $n$ is odd or $n$ is even but $c_1 \neq -(n+1)g$.

Here $c_1$ is the first Chern class of $X$ and $g$ is a generator of $H^2(X,Z)$ with the same sign of the Kahler class.

My question is :Has the case X is diffeomorphic to $\mathbb{CP}^n$ ,$n$ even and $c_1=-(n+1)g$ been ruled out in the following years?Or we have some constructions of manifolds of this type?