  We use recent data that show a narrow peak around $\sqrt{s}=2.37\ \text{GeV}$
in the $pn \to d \pi^+ \pi^-$ cross section, with about double strength at the
peak than in the analogous $pn \to d \pi^0 \pi^0$ reaction, and, assuming that
it is due to the excitation of a dibaryon resonance, we evaluate the cross
section for the $pn \to pn \pi^+ \pi^-$ reaction, with the final $pn$ unbound
but with the same quantum numbers as the deuteron. We use accurate techniques
to determine the final state interaction in the case of the $pn$ forming a
deuteron or a positive energy state, which allow us to get the $pn \to pn \pi^+
\pi^-$ cross section with $pn$ in I=0 and S=1, that turns out to be quite close
or saturates the experimental $pn \to pn \pi^+ \pi^-$ total cross section
around $\sqrt{s} = 2.37\ \text{GeV}$, depending on the angular momentum
assumed. We then parametrize a background with different methods, and the sum
of the resonant and background contributions is fitted to present data. The
resulting cross section exceeds the experimental results in the region of the
resonant peak, showing a problem in the dibaryon hypothesis. Yet, in view of
the dispersion of present experimental data, and the scarce information around
$\sqrt{s} = 2.37\ \text{GeV}$, a call is made for precise measurements of the
$pn \to pn\pi^+\pi^-$ reaction around this energy, to further clarify this
issue.
