Diffeomorphic Kähler manifolds with different Hodge numbers

This question was debated in another forum a few years ago. The result was a note by Frédéric Campana in which he describes a counterexample as a corollary of another construction. In 1986 Gang Xiao (An example of hyperelliptic surfaces with positive index Northeast. Math. J. 2 (1986), no. 3, 255–257.) found two simply connected complex surfaces $S$ and $S'$ (that is, complex dimension 2), with different Hodge numbers, that are homeomorphic by Freedman's classification. The homeomorphism has to be orientation-reversing, but $S \times S$ and $S' \times S'$ are orientedly diffeomorphic and of course still have different Hodge numbers. Freedman's difficult classification is not essential to the argument, because in 8 real dimensions you can use standard surgery theory to establish the diffeomorphism.

Campana also explains that Borel and Hirzebruch found the first counterexample in 1959, in 5 complex dimensions.

A few years after this post appeared, the question of oriented diffeomorphism invariance of Hodge and Chern numbers of smooth, projective varieties was settled completely by Kotschick and Schreieder in this paper. In fact, they even refer to this post(!).