Correlation between the luminosity of J = 2 ortho-H₂O⁺ and para-H₂O(2₁₁–2₀₂). The fitted function is L_H2O⁺∝L_H2O^α. We found a very good correlation between L_H2O⁺ and L_H2O with a slope close to one. Black points are from the local ULIRGs as listed in Table B.1. Dark blue ones are high-redshift starbursts from this work. Black solid lines indicate the χ² fitting results while the grey dashed lines and the grey annotations represent the average ratio between the L_H2O⁺ and L_H2O .