An embedding of the unit ball that does not embed into a Loewner chain

We construct a holomorphic embedding \(\phi :\mathbb B^3\rightarrow {\mathbb {C}}^3\) such that \(\phi ({\mathbb {B}}^3)\) is not Runge in any strictly larger domain. As a consequence, \(\mathcal S\ne {\mathcal {S}}^1\) for \(n=3\).