Palis' conjecture for surface diffeomorphisms

Let $M$ be a real compact surface, and let $k\geq 1$. For $f\in\mathrm{Diff}^k(M)$, a homoclinic tangency means a tangency between the stable and unstable manifolds of some saddle periodic point. Palis' conjecture. Every $f\in\mathrm{Diff}^k(M)$ can be $C^k$-approximated either by a hyperbolic diffeomorphism or by one exhibiting a homoclinic tangency. Since hyperbolic diffeomorphisms are structurally stable, this identifies homoclinic tangencies as a basic phenomenon responsible for bifurcations. The conjecture was proved for $k=1$ by E. Pujals and M. Sambarino, but remains open for $k>1$.