Context. Several studies have reported a correlation between the chemical abundances of stars and condensation temperature (known as Tc trend). Very recently, a strong Tc trend was reported for the ζ Reticuli binary system, which consists of two solar analogs. The observed trend in ζ2 Ret relative to its companion was explained by the presence of a debris disk around ζ2 Ret. Aims: Our goal is to re-evaluate the presence and variability of the Tc trend in the ζ Reticuli system and to understand the impact of the presence of the debris disk on a star. Methods: We used very high-quality spectra of the two stars retrieved from the HARPS archive to derive very precise stellar parameters and chemical abundances. We derived the stellar parameters with the classical (nondifferential) method, while we applied a differential line-by-line analysis to achieve the highest possible precision in abundances, which are fundamental to explore for very tiny differences in the abundances between the stars. Results: We confirm that the abundance difference between ζ2 Ret and ζ1 Ret shows a significant (~2σ) correlation with Tc. However, we also find that the Tc trends depend on the individual spectrum used (even if always of very high quality). In particular, we find significant but varying differences in the abundances of the same star from different individual high-quality spectra. Conclusions: Our results for the ζ Reticuli system show, for example, that nonphysical factors, such as the quality of spectra employed and errors that are not accounted for, can be at the root of the Tc trends for the case of individual spectra. Based on data obtained from the ESO Science Archive Facility under request number vadibekyan204818, vadibekyan204820, and vadibekyan185979.The tables with EWs of the lines and chemical abundances are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/591/A34