  Various theoretical calculations based on QCD or hadronic interactions
predict that in addition to the recently observed dibaryon resonance
$d^*(2380)$ with $I(J^P) = 0(3^+)$ there should also exist a dibaryon resonance
with mirrored quantum numbers $I(J^P) = 3(0^+)$. We report here on a search for
such a $NN$-decoupled state in data on the $pp \to pp\pi^+\pi^+\pi^-\pi^-$
reaction. Since no clear-cut evidence has been found, we give upper limits for
the production cross section of such a resonance in the mass range 2280 - 2500
MeV.
