Pujals–Sambarino-type description for codimension-one dominated splittings

Let $f$ be a $C^2$-diffeomorphism and let $K$ be a compact locally maximal invariant set of $f$ admitting a dominated splitting

$$TM|_K=E^s\oplus F,$$

where $F$ has dimension $1$ and $E^s$ is uniformly contracting. Codimension-one dominated-splitting conjecture. The set $K$ is the union of finitely many normally hyperbolic circles on which a power of $f$ is a rotation, periodic points contained in a union of finitely many normally hyperbolic periodic intervals, and finitely many pairwise disjoint homoclinic classes, each containing at most finitely many non-hyperbolic periodic points. The conjecture seeks to generalize the known two-dimensional description of non-wandering sets with dominated splittings; the source mentions progress when there is a unique non-hyperbolic periodic point, but leaves the general case open.