Herman's dominated-splitting conjecture for conservative diffeomorphisms

Let $M$ be a compact manifold and let $f\in \operatorname{Diff}^1_m(M)$ be a conservative diffeomorphism. Suppose there is a neighborhood $\mathcal{U}$ of $f$ in $\operatorname{Diff}^1_m(M)$ such that, for every $g\in\mathcal{U}$ and every periodic orbit $x$ of $g$, the matrix associated with the periodic orbit has at least one eigenvalue whose modulus differs from one. A dominated splitting is an invariant splitting of the tangent bundle satisfying the usual domination inequality.

Herman's conjecture. Under these assumptions, $f$ admits a dominated splitting.

The statement is presented as Herman's question relating robust exclusion of fully elliptic periodic orbits to dominated splittings. The supplied text gives no resolution status.