Let $X$ be an algebraic varietiy (as good as you want, say affine and smooth) and let us denote by $MHM(X)$ the category of mixed Hodge modules as descrived by Saito (see for example this or this).
Question: Given two algebraic varieties $X, Y$, what is the relation between $MHM(X), MHM(Y)$ and $MHM(X \times Y)$?
I don't believe that $MHM(X \times Y) = MHM(X) \,\boxtimes \,MHM(Y)$ holds (because variations of mixed Hodge structures with 'diagonal' support). But, what happens when we take the Grothendieck group of $MHM(X)$, $K(MHM(X))$? That is, the group generated by $[M]$ for $M \in MHM(X)$ with the relation $[M] = [M'] + [M'']$ if $0 \to M' \to M \to M'' \to 0$ is a short exact sequence. Do we have an isomorphism of abelian groups
$K(MHM(X \times Y)) \cong K(MHM(X)) \otimes K(MHM(Y))\,\,$?
If it wasn't true, is there any kind of relation between them, apart from the external product $\boxtimes: K(MHM(X)) \otimes K(MHM(Y)) \to K(MHM(X \times Y)$?
It it was true, does it also hold when considering the relative situation $K(MHM(X \times_Z Y)) \cong K(MHM(X)) \otimes_{K(MHM(Z))} K(MHM(Y))$ where $K(MHM(X)), K(MHM(Y))$ are endowed with a $K(MHM(Z))$-module structure by pullback of projections?
Thank you so much in advance!
