Abdenur–Bonatti–Díaz conjecture on non-wandering sets with nonempty interior

Let $M$ be the compact manifold underlying $Diff^1(M)$, let $Diff^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$, and let $\Omega(f)$ be the non-wandering set of $f$. A subset of $Diff^1(M)$ is residual if it contains a countable intersection of open dense subsets. Abdenur–Bonatti–Díaz conjecture. There exists a residual subset $R$ of $Diff^1(M)$ such that, for any $f \in R$, if $\Omega(f)$ has nonempty interior, then $f$ is transitive. The conjecture is false in the non-invertible setting, as stated in the source; the cited counterexample resolves the claim negatively in that setting.