The weakly hyperbolic toral-action conjecture

Let $\Gamma$ be as in the uniformly partially hyperbolic quasi-affine conjecture, and let $\mathbb T^n$ be the $n$-torus. A toral action is weakly hyperbolic when it has the weak hyperbolicity property used in the source. The weakly hyperbolic toral-action conjecture. If $\Gamma$ acts weakly hyperbolically on $\mathbb T^n$ while preserving a smooth measure, then the action should be affine. This weaker variant also remains open.