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Aims. We conducted a high-precision differential abundance analysis of the remarkable binary system HD 240429/30 (Krios and Kronos, respectively), whose difference in metallicity is one of the highest detected to date in systems with similar components (~0.20 dex). A condensation temperatureT_Ctrend study was performed to search for possible chemical signatures of planet formation. In addition, other potential scenarios are proposed to explain this disparity. Methods. Fundamental atmospheric parameters (T_eff, logg, [Fe/H],υ_turb) were calculated using the latest version of the FUNDPAR code in conjunction with ATLAS12 model atmospheres and the MOOG code, considering the Sun and then Kronos as references, employing high-resolution MAROON-X spectra. We applied a full line-by-line differential technique to measure the abundances of 26 elements in both stars with equivalent widths and spectral synthesis taking advantage of the non-solar-scaled opacities to achieve the highest precision. Results. We find a difference in metallicity of ~0.230 dex: Kronos is more metal rich than Krios. This result denotes a challenge for the chemical tagging method. The analysis encompassed the examination of the diffusion effect and primordial chemical differences, concluding that the observed chemical discrepancies in the binary system cannot be solely attributed to any of these processes. The results also show a noticeable excess of Li of approximately 0.56 dex in Kronos, and an enhancement of refractories with respect to Krios. A photometric study with TESS data was carried out, without finding any signal of possible transiting planets around the stars. Several potential planet formation scenarios were also explored to account for the observed excess in both metallicity and lithium in Kronos; none was definitively excluded. While planetary engulfment is a plausible explanation, considering the ingestion of an exceptionally high mass, approximately ~27.8M_⊕, no scenario is definitively ruled out. We emphasize the need for further investigations and refinements in modelling; indispensable for a comprehensive understanding of the intricate dynamics within the Krios and Kronos binary system.