Conjugacy separability of some factor groups of a free product

Groups of the form $F/C^{(n)}$ are studied, where $F$ is the free product of groups $B_i$, $i\in I$, and $C^{(n)}$ is the $n$th term of the derived series of the Cartesian subgroup of this product. It is proved that if every $B_i$ is conjugacy separable, residually finite with respect to occurrence in cyclic subgroups, and torsion-free, then the groups $F/C^{(n)}$ are conjugacy separable.
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