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The interaction between Λ baryons and kaons/antikaons is a crucial ingredient for the strangeness S=0 and S=−2 sector of the meson--baryon interaction at low energies. In particular, the ΛK¯¯¯¯ might help in understanding the origin of states such as the Ξ(1620), whose nature and properties are still under debate. Experimental data on Λ−K and Λ−K¯¯¯¯ systems are scarce, leading to large uncertainties and tension between the available theoretical predictions constrained by such data. In this Letter we present the measurements of Λ−K+⊕Λ¯¯¯¯−K− and Λ−K−⊕Λ¯¯¯¯−K+ correlations obtained in the high-multiplicity triggered data sample in pp collisions at s√=13 TeV recorded by ALICE at the LHC. The correlation function for both pairs is modeled using the Lednicky−Lyuboshits analytical formula and the corresponding scattering parameters are extracted. The Λ−K−⊕Λ¯¯¯¯−K+ correlations show the presence of several structures at relative momenta k∗ above 200 MeV/c, compatible with the Ω baryon, the Ξ(1690), and Ξ(1820) resonances decaying into Λ−K− pairs. The low k∗ region in the Λ−K−⊕Λ¯¯¯¯−K+ also exhibits the presence of the Ξ(1620) state, expected to strongly couple to the measured pair. The presented data allow to access the ΛK+ and ΛK− strong interaction with an unprecedented precision and deliver the first experimental observation of the Ξ(1620) decaying into ΛK−.