We study the spin-exotic ${J}^{PC}={1}^{\ensuremath{-}+}$ amplitude in single-diffractive dissociation of $190\text{ }\text{ }\mathrm{GeV}/c$ pions into ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ using a hydrogen target and confirm the ${\ensuremath{\pi}}_{1}(1600)\ensuremath{\rightarrow}\ensuremath{\rho}(770)\ensuremath{\pi}$ amplitude, which interferes with a nonresonant ${1}^{\ensuremath{-}+}$ amplitude. We demonstrate that conflicting conclusions from previous studies on these amplitudes can be attributed to different analysis models and different treatment of the dependence of the amplitudes on the squared four-momentum transfer and we thus reconcile these experimental findings. We study the nonresonant contributions to the ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ final state using pseudodata generated on the basis of a Deck model. Subjecting pseudodata and real data to the same partial-wave analysis, we find good agreement concerning the spectral shape and its dependence on the squared four-momentum transfer for the ${J}^{PC}={1}^{\ensuremath{-}+}$ amplitude and also for amplitudes with other ${J}^{PC}$ quantum numbers. We investigate for the first time the amplitude of the ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ subsystem with ${J}^{PC}={1}^{\ensuremath{-}\ensuremath{-}}$ in the $3\ensuremath{\pi}$ amplitude with ${J}^{PC}={1}^{\ensuremath{-}+}$ employing the novel freed-isobar analysis scheme. We reveal this ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ amplitude to be dominated by the $\ensuremath{\rho}(770)$ for both the ${\ensuremath{\pi}}_{1}(1600)$ and the nonresonant contribution. These findings largely confirm the underlying assumptions for the isobar model used in all previous partial-wave analyses addressing the ${J}^{PC}={1}^{\ensuremath{-}+}$ amplitude.
