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

>If $X$ is compact Kähler 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 $H^2(X,\mathbb{Z})$ with the same sign of the Kähler 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?