Classification problems of toric manifolds via topology

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We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.

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Toric cohomological rigidity of simple convex polytopes (2008, methodology): “In [10] the rigidity property is expressed in terms of toric manifolds, but here we modify the original definition to make use of a wider class of quasitoric manifolds.”
Cohomological rigidity of real Bott manifolds (2008, background): “We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold.”
Topological classification of generalized Bott towers (2008, background): “So we are led to ask how much information ordinary cohomology contains for toric manifolds and posed the following problem in [ 9 ].”
Rigidity problems in toric topology: A survey (2011, background): “Hence, they provide a lot of non-rigid simplicial polytopes combinatorially (even cohomologically, see [35]) but reducible.”
Toric Topology (2012, background): “For more results on the cohomological rigidity of (generalised) Bott towers, see the survey articles [211] and [76].”
Toric topology (2012, background): “Based on these observations, we posed the following problem in [64].”