The open amalgams conjecture for completely metrisable groups

A topological group $G$ is said to have property (OA) if, whenever $G=A*_CB$ is a non-trivial decomposition as a free product with amalgamation, the subgroups $A$, $B$, and $C$ are open in $G$. Here, non-trivial means that $A$ and $B$ are proper subgroups of $G$.

Open amalgams conjecture. Every completely metrisable topological group has property (OA).

Locally compact groups are known to have property (OA), and the paper proves the same for completely metrisable groups satisfying an additional somewhere-density condition. The conjecture proposes that this holds for all completely metrisable topological groups.