For this post, a group $G$ shall be referred to as generally divisible, in case $\forall{x\in G:}~\forall{n\in\mathbb{N}^{\times}:}~\exists{y\in G:}~y^{n}=x$.
Note. Here is no commutativity requirement on the group imposed!
Question. Let $\mathcal{D}$ be the collection of generally divisible, seperable metrisable topological groups. Is there $G\in\mathcal{D}$, such that for all $H\in\mathcal{D}$, there exists an injective homomorphism $f:H\to G$, which is open when restricted to its image? ${}^{\ast}$
($\ast$) i. e. the map $f: H\to f(H)$ is open, or equivalently, the inverse $f^{-1}:f(H)\to H$ is continuous.