Consider two measurable spaces $X$ and $Y$ and form their Cartesian product $X\times Y$, with the product $\sigma$-algebra.
In this question I asked whether, given probability measures $p$ and $q$ on $X\times Y$, their total variation distance
$$ d(p,q) = \sup_{f:X\times Y\to[0,1]} \left| \int f \, dp - \int f\, dq \right| $$ can be equivalently be tested using products of functions, i.e. whether $$ d(p,q) = \sup_{g:X\to[0,1],\, h:Y\to[0,1]} \left| \int gh \, dp - \int gh\, dq \right| . $$
The answer, in general, is negative.
However, what can we say if $p$ and $q$ are product measures, that is, if there are measures $m$ and $n$ on $X$ and $m',n'$ on $Y$ such that $p=m\otimes m'$ and $q=n\otimes n'$? Can we still find a counterexample in that case?
