A homotopy-theoretic rigidity property of Bott manifolds

The rigidity conjecture in toric topology posits that two toric manifolds are diffeomorphic if and only if their integral cohomology rings are isomorphic as graded rings. Only a few low dimensional cases have been resolved. We weaken the conjecture to one concerning homotopy type rather than diffeomorphism, and show that the weaker conjecture holds for Bott manifolds, once enough primes have been inverted. In particular, show that the rational homotopy type of a Bott manifold is determined by its rational cohomology ring.
The material in this paper was inspired by the mathematics discussed at the International conference «Toric Topology and Automorphic Functions» (September, 5–10th, 2011, Khabarovsk, Russia).