Berger–Rovella inverse-limit stability conjecture for endomorphisms

Let $f$ be an endomorphism, with inverse limit space $\overleftarrow M_f$ and natural extension $\overleftarrow f$. It is $C^1$-inverse limit stable if every $C^1$ perturbation $f'$ admits a homeomorphism $h:\overleftarrow M_f\to\overleftarrow M_{f'}$ such that $h\circ\overleftarrow f=\overleftarrow f'\circ h$. An axiom A endomorphism satisfies the weak transversality condition if, for every $\underline{x}\in\overleftarrow\Omega_f$ and every $y\in\Omega_f$, the map $\pi_0|W^u(\underline{x};\overleftarrow f)$ is transverse to $W^s_\epsilon(y;f)$. Berger–Rovella conjecture. The $C^1$-inverse limit stable endomorphisms are those which satisfy axiom A and the weak transversality condition. The source says that one direction was proved in the cited work and that the other direction remains open.