Conjecture For every totally bounded uniform space $(U;F)$ and filters $\mathcal{X}$, $\mathcal{Y}$ on $U$ such that $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$
there exist proper filters $\mathcal{X}'\supseteq\mathcal{X}$ and $\mathcal{Y}'\supseteq\mathcal{Y}$ such that $$\forall E\in F \exists X'\in\mathcal{X}', Y'\in\mathcal{Y}':X'\times Y'\subseteq E.$$