Preperiodicity conjecture for invariant measures

Let $f$ be a $C^r$ diffeomorphism and let $\mu$ be an invariant measure. The measure is $C^r$-preperiodic if it can be approximated, together with the diffeomorphism, by periodic measures of nearby $C^r$ diffeomorphisms.

Preperiodicity conjecture. For $r\geq 1$, there exists a $C^r$-residual set of diffeomorphisms for which any invariant measure supported in a chain-recurrent set is $C^r$-preperiodic.

This asks whether chain recurrence is sufficient for generic approximation of invariant measures by periodic measures under perturbation. The source notes that ergodic measures are always $C^1$-preperiodic, but gives no resolution for the stated claim.