Generic zero-exponents or minimal invariant foliation conjecture in dimension 3

Let $M^{3}$ be a three-dimensional smooth compact manifold, let $m$ be a smooth volume measure, and let $\operatorname{Diff}^{1}_{m}(M^{3})$ denote the volume-preserving $C^{1}$ diffeomorphisms. An invariant foliation is minimal if every leaf is dense, and it is expanding or contracting according to the corresponding dynamical behavior. Generic minimal-foliation conjecture. Generically in $\operatorname{Diff}^{1}_{m}(M^{3})$, either all Lyapunov exponents are zero almost everywhere, or else there exists a minimal invariant expanding or contracting foliation. The preceding theorem shows that such a foliation implies stable Bernoulliness; the conjecture concerns whether the alternative occurs generically in the nonzero-exponent case.