The Tameness Conjecture for train-track generic automorphisms

Let $M$ be a connected sum of copies of $S^2\times S^1$, with canonical splitting $M=H_1\cup H_2$, and let $f:M\to M$ be a train-track generic automorphism. A one-dimensional lamination in $M$ is tame if every point has a flat chart neighborhood in $M$.

Tameness Conjecture. The automorphism $f$ can be isotoped so that it preserves the splitting and the induced automorphism $g$ on $H_1$ is train-track generic and has a tame one-dimensional invariant lamination.

The conjecture concerns whether the invariant laminations associated with train-track generic automorphisms can be chosen without the pathological self-linking found in typical generic automorphisms. The supplied text does not state whether the conjecture has been resolved.