Ortho–para H₂ conversion reactions mediated by the exchange of a H⁺ proton have been investigated at very low energy for the first time by means of a time independent quantum mechanical (TIQM) approach. State-to-state probabilities and cross sections for H⁺ + H₂ (v = 0, j = 0,1) processes have been calculated for a collision energy, E_c, ranging between 10⁻⁶ eV and 0.1 eV. Differential cross sections (DCSs) for H⁺ + H₂ (v = 0, j = 1) → H⁺ + H₂ (v′ = 0, j′ = 0) for very low energies only start to develop a proper global minimum around the sideways scattering direction (θ ≈ 90°) at E_c = 10⁻³ eV. Rate coefficients, a crucial information required for astrophysical models, are provided between 10 K and 100 K. The relaxation ortho–para process j = 1 → j′ = 0 is found to be more efficient than the j = 0 → j′ = 1 conversion at low temperatures, in line with the extremely small ratio between the ortho and para species of molecular hydrogen predicted at the temperature of interstellar cold molecular clouds. The results obtained by means of a statistical quantum mechanical (SQM) model, which has previously proved to provide an adequate description of the dynamics of the title reactions at a higher collision energy regime, have been compared with the TIQM results. A reasonable good agreement has been found with the only exception of the DCSs for the H⁺ + H₂ (v = 0, j = 1) → H⁺ + H₂ (v′ = 0, j′ = 0) process at very low energy. SQM cross sections are also slightly below the quantum results. Estimates for the rate coefficients, in good accord with the TIQM values, are a clear improvement with respect to pioneering statistical studies on the reaction.