$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_n^{\oplus d}]$ be the polynomial ring on $dn^2$ variables corresponding to the coordinates of $d$ $n\times n$ matrices. Let these matrix variables be $X_1,\ldots,X_d$. For an $n\times n$ matrix $A$, let $c_k(A)$ be the coefficient of $T^k$ in the characteristic polynomial $\det(A-TI)$.

The group scheme $\GL_{n,R}$ acts on $R[n]$ by simultaneous conjugation on the $d$ matrices. Clearly for any product $X_{i_1}X_{i_2}\cdots X_{i_r}$ (where $i_j\in[1\ldots d], r\ge 1$), the function $c_k(X_{i_1}\cdots X_{i_r})\in R[n]$ is invariant under $\GL_{n,R}$.

Is it true that for any commutative ring $R$, $R[n]^{\GL_{n,R}}$ is generated as an $R$-algebra by the functions $c_k(X_{i_1}\cdots X_{i_r})$?

Does anyone have a reference for this?

Remark - The statement over $\mathbb{C}$ is a classical result of Sibirski and Procesi. This was later extended to the case $R = \mathbb{Z}$ and $R$ any algebraically closed field by Donkin in Invariants of several matrices. In Concini-Procesi's The invariant theory of matrices, they also seem to obtain the result when $R$ is an infinite field.