ag.algebraic geometry - Permanence of regularity in "generalised" semistable models

Given a regular ring $A$ with an element $t$, consider the "generalised" semistable model
$S := A[X_1,...,X_n]/(P_1cdots P_n - t)$ over $A$, where $P_i := {X_i}^{e_i}$ and $e_i$ are positive integers.

Question: For which $A$, $t$, and $(e_i)$ is $S$ regular?

E.g., when $n=2$ and $A$ is a discrete valuation ring and $t$ is a uniformiser, then it is easy to prove that $S$ is regular when $e_1$ and $e_2$ are both equal to 1. When $nge 3$ and $(A,t)$ is a DVR with uniformiser $t$, is there a reference?