There are three conjectures on group rings that bear the name of Kaplansky (see for example [this question][1]). The `unit conjecture' in the title of the present question is the stronger among them, and states that the group ring $\mathbb{C}\Gamma$ of a torsion-free group $\Gamma$ should contain no units besides the obvious ones $\lambda g$ for $\lambda\in\mathbb{C}^\times,\, g\in\Gamma$. 

A natural combinatorial property on a group $\Gamma$ under which the conclusion of the unit conjecture is known to hold is the unique products property: one says that $\Gamma$ has unique products if for any two finite (nonempty) subsets $A,B$ in $\Gamma$ there exists $a\in A,b\in B$ such that $ab\not= a'b'$ for all $(a,b)\not= (a',b')\in A\times B$ (informally, $ab$ can be written in only one way as a product). This property has been well-studied, and is known to hold for various classes of groups; it is also known that there are torsion-free groups which have non-unique products (see for example [this paper][2] of B. Bowditch for further references).

While the other two conjectures can be approached by a variety of means (see the afore-mentioned MO question for more details), I am not aware of any torsion-free group $\Gamma$ for which the conclusion uf the unit conjecture is known to hold, without the unique products property having been established first. 

Hence (at last) my query: is there a known example of a (say finitely generated) torsion-free group $\Gamma$ such that it is known that all units in $\mathbb{C}\Gamma$ are the obvious ones, but for which it is not known that it has unique products (or even better, such that it is known to have non-unique products)? 


  [1]: http://mathoverflow.net/questions/79559/what-is-the-current-status-of-the-kaplansky-zero-divisor-conjecture-for-group-ri
  [2]: http://www.ams.org/mathscinet-getitem?mr=1794287