Motivated by the persistent anomalies in the semileptonic $B$-meson decays, we investigate the competency of LHC data to constrain the $R_{D^{(*)}}$-favoured parameter space in a charge $-1/3$ scalar leptoquark ($mathcal S_1$) model. We consider some scenarios with one large free coupling to accommodate the $R_{D^{(*)}}$ anomalies. As a result, some of them dominantly yield nonresonant $tautau$ and $tau u$ events at the LHC through the $t$-channel $mathcal S_1$ exchange. So far, no experiment has searched for leptoquarks using these signatures and the relevant resonant leptoquark searches are yet to put any strong exclusion limit on the parameter space. We recast the latest $tautau$ and $tau u$ resonance search data to obtain new exclusion limits. The nonresonant processes strongly interfere (destructively in our case) with the Standard Model background and play the determining role in setting the exclusion limits. To obtain precise limits, we include non-negligible effects coming from the subdominant (resonant) pair and inclusive single leptoquark productions systematically in our analysis. To deal with large destructive interference, we make use of the transverse mass distributions from the experiments in our statistical analysis. In addition, we also recast the relevant direct search results to obtain the most stringent collider bounds on these scenarios to date. These are independent bounds and are competitive to other known bounds. Finally, we indicate how one can adopt these bounds to a wide class of models with $mathcal S_1$ that are proposed to accommodate the $R_{D^{(*)}}$ anomalies.
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