Finiteness conjecture for diffeomorphism types of real forms

Let $M(\mathbb C)$ be a smooth projective manifold. A real form of $M(\mathbb C)$ is a real algebraic manifold whose complexification is $M(\mathbb C)$. Finiteness conjecture. The set of diffeomorphism types of real forms of $M(\mathbb C)$ is finite. This is presented as one of several conjectures about real forms; the source gives no precise definition of “simple” and no resolution of this finiteness question.