  We examine various scenarios in which the Standard Model is extended by a
light leptoquark state to explain one or both $B$-physics anomalies. Combining
low-energy constraints and direct searches at the LHC, we confirm that the only
single leptoquark model that can explain both anomalies at the same time is a
vector leptoquark, known as $U_1$. Focusing on $U_1$, we highlight the
complementarity between LHC and low--energy constraints, and argue that
improving the experimental bound on $\mathcal{B}(B\to K\mu\tau)$ by two orders
of magnitude could compromise its viability as a solution to the $B$-physics
anomalies.

High Energy Physics - Phenomenology