In a book I am reading, "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents" by Luc Illusie, Yves Laszlo, Fabrice Orgogozo (https://arxiv.org/abs/1207.3648), there is the following assertion whose proof I would like to understand.

Let $e_1, ..., e_n$ be natural numbers, not all zero. For any trait $S$ with uniformizer $\pi$, define $$ V(S,\pi,e_1,...,e_n) = Spec(O_S[T_1,\ldots,T_n]/(T_1^{e_1}...T_n^{e_n}-\pi)). $$ It is claimed that this scheme is regular in Exposé XVI, Lemma 3.5.13. However the proof is only briefly outlined. If someone can provide details, I would appreciate it.

The first case is if at least one of $e_i$ are "invertible in $\eta$" (generic point of $S$), which, I don't even see why that would be relevant, but this case is supposed to be an easy exercise. Is there a regularity criterion which would be easy to apply here?