Cohomological rigidity conjecture for toric Fano manifolds

Let $X$ and $Y$ be toric Fano manifolds, and let $H^*(X)$ and $H^*(Y)$ denote their cohomology rings. A cohomology ring isomorphism preserves their first Chern classes when it maps $c_1(X)$ to $c_1(Y)$. Cohomological rigidity conjecture. If there is a cohomology ring isomorphism between $X$ and $Y$ that preserves their first Chern classes, then $X$ and $Y$ are isomorphic as varieties. This conjecture is motivated by the question of uniqueness of toric actions on monotone symplectic manifolds and has been verified for Fano Bott manifolds, but remains open for toric Fano manifolds in general.