The affine-action classification conjecture

Let $\Gamma<SL(n,\mathbb R)$ be a lattice. An action of $\Gamma$ on a compact manifold is affine-connection preserving when it preserves both a volume form and an affine connection. The affine-action classification conjecture. Actions of $\Gamma$ with these properties should be classifiable, and every such action should be algebraically defined in the sense specified in the source. This is a geometric rigidity conjecture generalizing the expected algebraic description of standard examples such as the action on a torus.