Let $3\leq k\leq 8$ be an integer. Suppose $M$ is a complex surface which has a Kahler-Einstein metric and has the same Betti numbers as $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, i.e. $$b_0(M)=b_4(M)=1,~~ b_1(M)=b_3(M)=0,~~b_2^+(M)=1,~~ b_2^-(M)=k.$$ Then can we conclude that $M$ is biholomorphic to $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$?

**Note added**: The answer of Martin de Borbon has shown that this is not true for $5\leq k\leq 8$. I would like to know if the statement is true for $k=3, 4$.