  This paper reports the first T2K measurement of the transverse kinematic
imbalance in the single-$\pi^+$ production channel of neutrino interactions. We
measure the differential cross sections in the muon-neutrino charged-current
interaction on hydrocarbon with a single $\pi^+$ and at least one proton in the
final state, at the ND280 off-axis near detector of the T2K experiment. The
extracted cross sections are compared to the predictions from different
neutrino-nucleus interaction event generators. Overall, the results show a
preference for models which have a more realistic treatment of nuclear medium
effects including the initial nuclear state and final-state interactions.
