Higher-dimensional almost homoclinic sequence conjecture

Let $f:M\rightarrow M$ be a $\mathrm{C}^{2}$ diffeomorphism with a compact basic set $\Lambda$ which has associated an almost homoclinic sequence. Let $p\in\Lambda$ be a hyperbolic periodic point. Almost homoclinic sequence conjecture. One of the following statements holds:

1. $p$ has an associated transversal homoclinic point outside $\Lambda$.
2. For every neighborhood $\mathcal{N}\subset\operatorname{Diff}^{1}(M)$ of $f$, there exists $g\in\mathcal{N}$ having a homoclinic bifurcation associated to $p$.

This extends the corresponding surface result to higher dimensions, where the authors note that examples of Díaz show the preceding theorem is false as stated. The conjectured alternative replaces the surface conclusion involving homoclinic tangencies by the broader notion of a homoclinic bifurcation.