Conjecture on the central bundle of aperiodic classes

Let $\Lambda$ be an aperiodic class for a $C^1$-generic diffeomorphism $f$, and let

$$E^s\oplus E^c\oplus E^u$$

be a dominated splitting such that $E^s$ and $E^u$ are respectively the maximal uniformly contracted and expanded sub-bundles. Conjecture on the central bundle of aperiodic classes. The bundle $E^c$ has dimension at least $2$ and does not admit a finer dominated splitting. The conjecture is part of a proposed description of aperiodic classes with dominated splittings; the source gives no resolution.