  The measurements of $R_K^{(*)}$ and $R_{D}^{(*)}$ by BaBar, Belle and the
LHCb collaborations could be showing a hint of lepton flavor universality
violation that can be accommodated by the presence of suitable leptoquarks at
the TeV scale. We consider an effective description, with leptoquarks arising
as composite pseudo Nambu-Goldstone bosons, as well as anarchic partial
compositeness of the SM fermions. Considering the $R_K^{(*)}$ anomaly within
this framework, we study pair production of $S_3\sim(\bar 3,3)_{1/3}$ at the
LHC. We focus on the component $S_3^{1/3}$ of the triplet, which decays
predominantly into $t\tau$ and $b\nu$, and study the bounds from existing
searches at $\sqrt{s}=13$ TeV at the LHC. We find that sbottom searches in the
$b\bar{b}$+MET final state best explore the region in parameter space preferred
by our model and currently exclude $S_3^{1/3}$ masses up to $\sim$1 TeV.
Additional searches, considering the $t\tau$ and $t\mu$ decay modes, are
required to probe the full physical parameter space. In this paper we also
recast existing studies on direct leptoquark searches in the $t\tau t\tau$
channel and SM $t\bar{t}t\bar{t}$ searches, and obtain the regions in parameter
space currently excluded. Practically the whole physical parameter space is
currently excluded for masses up to $\sim$0.8 TeV, which could be extended up
to $\sim$1 TeV with the full Run 3 dataset. We conclude that pair production
searches for this leptoquark can benefit from considering the final state $t
\tau b$ +MET, where the largest branching ratio is expected. We appraise that
future explorations of leptoquarks explaining the B-anomalies with masses
beyond the TeV should also consider single and non-resonant production in order
to extend the mass reach.

High Energy Physics - Phenomenology