Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle $\lbrace |z| = r \rbrace \subset \mathbb C^\times$ contains at most two points of the collection.
There are some obvious generators of the 1-cohomology of such the configuration space, like those represented by $d \log z_i$, $d \log (z_i - z_j)$, and pull-backs of the generators under the moment maps $m_{ijk}\colon (z_1,\ldots,z_n) \mapsto (|z_i|,|z_j|,|z_k|)$. Is a complete description of the cohomology ring of $Conf_{1,n}^3$ in terms of generators and relations known?
