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We examine various scenarios in which the Standard Model is extended by a light leptoquark state to solve for one or both B-physics anomalies, viz. R_D(◦)^exp > R_D(◦)^SM or/and R_K(◦)^exp < R_K(◦)^SM. To do so we combine the constraints arising both from the low-energy observables and from direct searches at the LHC. We find that none of the scalar leptoquarks of mass mLQ ≃ 1 TeV can alone accommodate the above mentioned anomalies. The only single leptoquark scenario which can provide a viable solution for mLQ ≃ 1÷2 TeV is a vector leptoquark, known as U1, which we re-examine in its minimal form (letting only left-handed couplings to have non-zero values). We find that the limits deduced from direct searches are complementary to the low-energy physics constraints. In particular, we find a rather stable lower bound on the lepton flavor violating b → sℓ_1^±ℓ_2^± modes, such as B(B → K μΤ). Improving the experimental upper bound on B(B → K μΤ) by two orders of magnitude could compromise the viability of the minimal U1 model as well.