Data accumulated recently for the exclusive measurement of the $ppto pppi^+pi^-$ reaction at a beam energy of 0.793 GeV using the COSY-TOF spectrometer have been analyzed with respect to possible events from the $pp to nnpi^+pi^+$ reaction channel. The latter is expected to be the only $pipi$ production channel, which contains no major contributions from resonance excitation close to threshold and hence should be a good testing ground for chiral dynamics in the $pipi$ production process. No single event has been found, which meets all conditions for being a candidate for the $pp to nn pi^+pi^+$ reaction. This gives an upper limit for the cross section of 0.16 $mu$b (90% C.L.), which is more than an order of magnitude smaller than the cross sections of the other two-pion production channels at the same incident energy.
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