Generalized quasi-affine action conjecture under rigidity or dominated splitting

Let $G$ be a higher-rank simple Lie group, let $\Gamma<G$ be a lattice, let $M$ be a compact manifold, and let $\rho:\Gamma\to\operatorname{Diff}(M)$ be an action. Assume that the action preserves a rigid geometric structure or that a single element admits a dominated splitting. Generalized quasi-affine action conjecture. Then the action is generalized quasi-affine. This is proposed as a broader variant of the stiff-actions conjecture; the source notes that existing non-generalized-quasi-affine constructions involve singular divisors carrying non-invariant stationary measures, but gives no resolution of this claim.