In the paper *On the complex projective spaces*, Hirzebruch and Kodaira prove the following:

If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ if $n$ 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 as the Kähler class.

My question is: Has the case $X$ is diffeomorphic but not biholomorphic to $\mathbb{CP}^n$ with $n$ even and $c_1=-(n+1)g$ been ruled out in the following years? Or do we have some constructions of manifolds of this type?