Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

It is well-known, that the ring $\mathbb{C} [X_1,\dotsc,X_n]^{A_n}$ of polynomial invariants of the alternating group $A_n$ has no finite SAGBI basis with respect to the lexicographical order for any number of variables $n \ge 3$. This note proves the existence of a nonsingular matrix $\delta_n \in GL(n,\mathbb{C} )$ such that the ring of polynomial invariants $\mathbb{C} [X_1,\dotsc,X_n]^{A_n^{\delta_n}}$, where $A_n^{\delta_n}$ denotes the conjugate of $A_n$ with respect to $\delta_n$, has a finite SAGBI basis for any $n \geq 3$.