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We present the first threefold differential measurement for neutral-pion multiplicity ratios produced in semi-inclusive deep-inelastic electron scattering on carbon, iron, and lead nuclei normalized to deuterium from CLAS at Jefferson Lab. We found that the neutral-pion multiplicity ratio is maximally suppressed for the leading hadrons (energy fraction $z\to$1), suppression varying from 25% in carbon up to 75% in lead. An enhancement of the multiplicity ratio at low $z$and high $p_T^2$is observed, suggesting an interconnection between these two variables. This behavior is qualitatively similar to the previous twofold differential measurement of charged pions by the HERMES Collaboration and, recently, by CLAS Collaboration. The largest enhancement was observed at high $p_T^2$for heavier nuclei, namely, iron and lead, while the smallest enhancement was observed for the lightest nucleus, carbon. This behavior suggests a competition between partonic multiple scattering, which causes enhancement, and hadronic inelastic scattering, which causes suppression.