The groups $G_{k+1}^k$ and fundamental groups of configuration spaces

In \cite{HigherGnk}, the author has constructed natural maps from fundamental groups of topological spaces (restricted configuration spaces) to the groups $G_n^k$. In the present paper, we show that in the case of $n=k+1$, the group $G_{k+1}^k$ is isomorphic to the fundamental group of some (quotient space of) some configuration space. In particular, this leads to the solution of word and conjugacy problems in $G_4^3$ and sheds light on $G_{k+1}^k$ for higher $k$.