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The high-flux test module (HFTM) of the International Fusion Materials Irradiation Facility (IFMIF) will be exposed to an intense flux of fast neutrons with energy values of up to approximately 55 MeV. The threshold of tritium-producing reactions in the HFTM and the specimen materials is typically between 10 and 20 MeV: therefore, the assessment of the tritium inventory becomes important for the handling of the specimen cells and specimens after the irradiation cycle. In this paper, we calculate the tritium production in the specimen cells by neutrons with the McDeLicious code and activation nuclear data from the libraries EAF-2007 and IEAF-2001. The results differ by roughly a factor of two. We discuss ambiguities in the relevant activation data and conclude that better experimental verification of cross sections of isotopes most important for the tritium production in the test cells is desirable.