The classification conjecture for irreducible higher-rank Anosov actions

Let a higher-rank Anosov action mean an action of a higher-rank Abelian group with at least one normally hyperbolic element, and call such an action irreducible when it has no nontrivial algebraic or dynamical factor as intended in the classification setting. The actions considered are higher-rank ${\mathbb Z}^k$- and ${\mathbb R}^k$-actions on compact manifolds.

Classification conjecture. Every irreducible higher-rank ${\mathbb Z}^k$- or ${\mathbb R}^k$-Anosov action on any compact manifold is smoothly conjugate to an algebraic action.

Known examples are algebraic or reducible, and several local and global rigidity results support the conjecture, but the classification remains open in general.