  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.
$R_{D^{(\ast)}}^\mathrm{exp}> R_{D^{(\ast)}}^\mathrm{SM}$ or/and
$R_{K^{(\ast)}}^\mathrm{exp}< R_{K^{(\ast)}}^\mathrm{SM}$. 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
$m_\mathrm{LQ} \simeq 1$ TeV can alone accommodate the above mentioned
anomalies. The only single leptoquark scenario which can provide a viable
solution for $m_\mathrm{LQ} \simeq 1\div 2$ TeV is a vector leptoquark, known
as $U_1$, 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
$b\to s\ell_1^\pm\ell_2^\mp$ modes, such as $\mathcal{B}(B\to K\mu\tau)$.
Improving the experimental upper bound on $\mathcal{B}(B\to K\mu\tau)$ by two
orders of magnitude could compromise the viability of the minimal $U_1$ model
as well.
