Generic positive exponent conjecture for random products in higher dimensions

Let $S$ be a manifold of dimension greater than $2$. A random product is the random dynamical system generated by the volume-preserving diffeomorphisms under consideration.

Higher-dimensional random-product conjecture. There exist a $C^1$ dense and $C^r$ open set of volume-preserving diffeomorphisms such that the random product has at least one positive exponent.

This conjecture proposes a higher-dimensional analogue of the paper's positive-exponent result. The authors explain that their perturbative argument may extend beyond two-dimensional fibers in the random-product setting because the perturbations are locally constant, but the claim is presented as an open question.