Pestov's conjecture on the universal ambit and enveloping semigroup

Let $G$ be a topological group. The universal ambit $S(G)$ and the enveloping semigroup $E(M(G))$ are associated by the canonical map

$$\varphi:S(G)\rightarrow E(M(G)).$$

Pestov's conjecture. The canonical map is an isomorphism if and only if $G$ is precompact. This conjecture concerns when the two canonical dynamical objects associated with a topological group coincide; the source describes it as arising from the diversity of counterexamples to Ellis's question. It has been disproved.