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The temperature dependence of the two superconducting gaps in pressurised <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>3</mn></msub><mi>S</mi></mrow></semantics></math></inline-formula> at 155 GPa with a critical temperature of 203 K has been determined using a data analysis of the experimental curve of the upper critical magnetic field as a function of temperature in the framework of the two-band <i>s</i>-wave Eliashberg theory. Two different phonon-mediated intra-band Cooper pairing channels in a regime of moderate strong couplings have the key role of the pair-exchange interaction between the two gaps, giving the two non-diagonal terms of the coupling tensor, which are missing in the single-band s-wave Eliashberg theory. The results provide a prediction of the different temperature dependence of the small and large gaps as a function of temperature, which provides evidence of multigap superconductivity in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>3</mn></msub><mi>S</mi></mrow></semantics></math></inline-formula>.