We monitor the phonon sideband emission from paraexcitons confined in a strain trap in cuprous oxide at T=1.25 K. On the low energy ank of the optical phonon replicas, both of Gamma₅^- and Gamma₃^- symmetry (the latter activated by application of a magnetic field), we detect sharp peaks that might represent indications for a paraexciton Bose-Einstein condensate. In contrast, such peaks are absent in the phonon-mediated emission of the orthoexcitons, and they also disappear at elevated temperatures. The results challenge our understanding of the involved physics, e. g., of the Auger recombination of excitons, which has so far been believed to prevent crossing the border to a condensate. We gratefully acknowledge the support of this project by the Deutsche Forschungsgemeinschaft in the frame of the ICRC TRR 160 (project A1). MB also acknowledges support by the RF Government Grant No. 14.Z50.31.0021.

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