I'm trying to show that any minimal WAP dynamical system $(X, G)$ is almost periodic. By Ellis's joint continuity theorem, it suffices to show that any minimal WAP system is distal. There are many references that prove this under the additional assumption that $X$ is metrizable.
I feel like there should be some slick enveloping semigroup proof of this. For instance if $G$ is abelian, we can do the following. Say $x\neq y\in X$ are proximal. Then there is some $p\in E(X)$ with $xp = yp$. We may suppose that $p$ is a minimal idempotent, so that $Xp$ is a proper subset of $X$. By WAP, $p$ is continuous, so $Xp\subsetneq X$ is compact, and as $G$ is abelian, $Xp$ is a subflow of $X$, contradicting minimality.