For sets $\cal X$ and $\cal Y$, let $a:{\cal X}\times{\cal X}\rightarrow \mathbb{R}$ and $a:{\cal Y}\times{\cal Y}\rightarrow \mathbb{R}$ be positive definite symmetric kernels. Define the tensor product $a\otimes b$ as usual via $$ (a\otimes b)((x,x'),(y,y')) = a(x,x')a(y,y') $$ for all $x,x'\in \cal X$ and $y,y'\in \cal Y$.

A positive definite kernel defines both a Gaussian process (GP) and a reproducing kernel Hilbert space (RKHS), also known as the Cameron-Martin space of the GP (and defined as the set of shift vectors for the GP that preserve equivalence).

Define ${\cal R}_k$ as the RKHS with reproducing kernel $k$. Defining $$ {\cal R}_a\otimes{\cal R}_b = \langle\{f\otimes g|f\in{\cal R}_a,g\in{\cal R}_b,\}\rangle ,$$ the vector product tensor product, it is well-known that $$ {\cal R}_{a\otimes b} \subset {\cal R}_a\otimes{\cal R}_b $$ where the subset is strict in the infinite dimensional case, and \begin{equation}\tag{1}\label{closure} {\cal R}_{a\otimes b} = \overline{{\cal R}_a\otimes{\cal R}_b} \end{equation} where the closure is with respect to the tensor product norm.

My question is what can we say about the supports of the corresponding Gaussian processes? Denoting an appropriately defined support of a Gaussian process $GP(0,a)$ by ${\cal S}_a$, is it true that $$ {\cal S}_{a\otimes b} \subset {\cal S}_a\otimes{\cal S}_b $$ and if so is it possible to define some kind of closure operation so that an analogue of \eqref{closure} holds?

Background: GPs with tensor product kernels are used a lot in GP regression, so an answer could help to understand why it may make sense to use GPs with tensor product kernels.