Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)? 


Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety? 



Moishezon manifold vs proper complex variety
**Edit:** I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.


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The only example I am aware of is due to Oguiso, *[Two remarks on Calabi-Yau Moishezon threefolds](https://doi.org/10.1515/crll.1994.452.153)*, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = \mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.


