Persistence conjecture for compact normally AS laminations

A compact lamination $(L,\mathcal L)$ embedded in a manifold $M$ is preserved by a diffeomorphism $f$ and is normally $AS$ if there exists $\epsilon>0$ such that the saturated nonwandering set of $f$ restricted to $L$ is normally hyperbolic and plaque-expansive, the $\epsilon$-local stable set of each leaf is transverse to the $\epsilon$-local unstable set of every other leaf with intersection contained in $L$, and the saturated nonwandering set is locally maximal in $M$. Persistence conjecture. Every compact normally $AS$ lamination is $C^1$-persistent. This conjecture is proposed as a common generalization of structural stability in dimension two and the Hirsch–Pugh–Shub persistence theorem; the supplied text gives no resolution, so its status remains open.