Cohomological rigidity problems in toric topology

Classification of compact smooth toric varieties (which we call toric manifolds) as varieties reduces to classification of their fans as is well-known. However, not much is known for classification of toric manifolds as smooth manifolds. One interesting and naive question is Cohomological rigidity problem for toric manifolds [3]. Are two toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic as graded rings?
Some partial affirmative solution and no counterexample is known to the problem so far. Similar questions can be asked for polytopes [1], real toric manifolds [2] and symplectic toric manifolds [4]. In this talk I will discuss these problems.
References
[1] S. Choi, T. Panov and D. Y. Suh, Toric cohomological rigidity of simple convex polytopes, arXiv: 0807.4800, to appear in Jour. London Math. Soc.
[2] Y. Kamishima and M. Masuda, Cohomological rigidity of real Bott manifolds, Algebraic & Geometric Topology 9 (2009) 2479–2502.
[3] M. Masuda and D. Y. Suh, Classification problems of toric manifolds via topology, Toric Topology, Contemp. Math. 460 (2008), 273–286; arXiv: 0709.4579.
[4] D. McDuff, The topology of toric symplectic manifolds, arXiv: 1004.3227.