For every integer k⩾1, we construct a torsion-free hyperbolic group without unique product all of whose subgroups up to index k are themselves non-unique product groups. This is achieved by generalizing a construction of Comerford to graphical small cancellation presentations, showing that for every subgroup H of a graphical small cancellation group there exists a free group F such that H*F admits a graphical small cancellation presentation.
Gruber, D., Martin, A., & Steenbock, M. (2015). Finite index subgroups without unique product in graphical small cancellation groups. Bulletin of the London Mathematical Society, 47(4), 631–638. https://doi.org/10.1112/blms/bdv040