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During the last years ATLAS and CMS have reported a number of slight to mild discrepancies in signatures of multileptons plus $b$-jets in analyses such as $t\bar t H$, $t\bar t W^\pm$, $t\bar t Z$ and $t\bar t t\bar t$. Among them, a recent ATLAS result on $t\bar t H$ production has also reported an excess in the charge asymmetry in the same-sign dilepton channel with two or more $b$-tagged jets. Motivated by these tantalizing discrepancies, we study a phenomenological New Physics model consisting of a $Z'$ boson that couples to up-type quarks via right-handed currents: $t_Rγ^μ\bar t_R$, $t_Rγ^μ\bar c_R$, and $t_R γ^μ\bar u_R$. The latter vertex allows to translate the charge asymmetry at the LHC initial state protons to a final state with top quarks which, decaying to a positive lepton and a $b$-jet, provides a crucial contribution to some of the observed discrepancies. Through an analysis at a detector level, we select the region in parameter space of our model that best reproduces the data in the aforementioned $t\bar t H$ study, and in a recent ATLAS $t\bar t t \bar t$ search. We find that our model provides a better fit to the experimental data than the Standard Model for a New Physics scale of approximately $\sim$500 GeV, and with a hierarchical coupling of the $Z'$ boson that favours the top quark and the presence of FCNC currents. In order to estimate the LHC sensitivity to this signal, we design a broadband search featuring many kinematic regions with different signal-to-background ratio, and perform a global analysis. We also define signal-enhanced regions and study observables that could further distinguish signal from background. We find that the region in parameter space of our model that best fits the analysed data could be probed with a significance exceeding 3 standard deviations with just the full Run-2 dataset.