The minimal expanding-or-contracting foliation conjecture in dimension three

Let $M^3$ be a three-dimensional manifold, let $m$ denote Lebesgue measure, and let $\operatorname{Diff}^{1}_{m}(M^3)$ be the space of $C^1$ Lebesgue-measure-preserving diffeomorphisms. A foliation is minimal if every leaf is dense, and an invariant expanding or contracting foliation is one arising from an invariant expanding or contracting one-dimensional subbundle. Minimal foliation conjecture. Generically in $\operatorname{Diff}^{1}_{m}(M^3)$, if $f$ has positive metric entropy, then there exists a minimal invariant expanding or contracting foliation. The source presents this as a possible approach to the positive-entropy stable ergodicity conjecture; it remains open.