$\DeclareMathOperator\GK{GK}$Let $k$ be the base field.
The Gelfand-Kirillov dimension was introduced by Gelfand and Kirillov in their seminal paper on the Gelfand-Kirillov conjecture.
A very famous paper that brought a lot of attention to the subject is Borho and Kraft 'Über die Gelfand-Kirillov Dimension', and for some time the paper was used a lot, until it was superseded by new presentations of the subject – such as Krause and Lenegan book on GK-dimension.
Eventually, people discovered that some results in Borho and Kraft's paper were false. One of their statement was: let $A$ and $B$ be two affine algebras, and consider their tensor product as an algebra. Then $$\GK( A \otimes_k B) = \GK ( A )+ \GK (B).$$
Albeit being an extremely natural thing to expect, this result is false, as was discovered by Warfield. He also provided a very simple technical assumption that makes that result true:
(Warfield) Consider a finite dimensional vector space $V$ which contains $1$ and generates $A$ as an algebra. Let $d_V(n)=\dim_k V^n$ where inductively, $V^0=k, V^n=V. V^{n-1}$ Suppose $\GK(A)=\lim_{n \to \infty} \log_n d_V(n)$. Then $$\GK( A \otimes_k B) = \GK ( A )+ \GK (B).$$
The above condition holds for almost commutative algebras, Noetherian PI-algebras, etc.
Now to my question proper.
Let $A$ and $B$ be two affine algebras, $M$ a f.g. left $A$-module, $N$ a f.g. left $B$-module. We can turn $M \otimes_k N$ into an $D:= A \otimes_k B$-module.
Is $\GK_D(M \otimes N)=\GK_A(M)+\GK_B(N)$? I know this is true, for instance, for holonomic modules for the Weyl algebras. But what can be said in general?
