Families of degenerating Poincar\'e-Einstein metrics on $\mathbb{R}^4$

We provide the first example of continuous families of Poincar\'e-Einstein metrics developing cusps on the trivial topology $\mathbb{R}^4$. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Pleba\'nski-Demia\'nski. We additionally indicate how to construct similar examples on more complicated topologies.