low-energies by the following effective Lagrangian, at dimension-6, $$\begin{aligned}
\label{leff:bc}
\mathcal{L}_{\mathrm{eff}} = -2 \sqrt{2}& G_F V_{cb} \bigg{[}(1+g_{V_L}^{\ell})\,\big{(}\bar{c}_L \gamma^\mu b_L\big{)}\big{(}\bar{\ell}_{L} \gamma_\mu \nu_{L}\big{)} +g_{V_R}^{\ell}\,\big{(}\bar{c}_R \gamma^\mu b_R\big{)}\big{(}\bar{\ell}_{L} \gamma_\mu \nu_{L}\big{)}\\
& + g_{S_R}^{\ell}\,\big{(}\bar{c}_L  b_R\big{)}\big{(}\bar{\ell}_{R} \nu_{L}\big{)}+ g_{S_L}^{\ell}\,\big{(}\bar{c}_R b_L\big{)}\big{(}\bar{\ell}_{R} \nu_{L}\big{)}+ g_T^{\ell}\, \big{(}\bar{c}_R \sigma^{\mu\nu} b_L\big{)}\big{(}\bar{\ell}_{R} \sigma_{\mu\nu} \nu_{L}\big{)}\bigg{]}+\mathrm{h.c.,}\nonumber\end{aligned}$$ where $g_{V_{L(R)}}^\ell$, $g_{S_{L(R)}}^\ell$ and $g_T^\ell$ are generic Wilson coefficients. Since the NP responsible for this interaction must arise above the electro-weak scale, one should further impose $SU(2)_L\times U(1)_Y$ gauge invariance to the above Lagrangian [@Buchmuller:1985jz; @Grzadkowski:2010es]. The operators one should then consider are [@Aebischer:2015fzz] $$\begin{aligned}
\label{eq:su2-operators}
\begin{split}
\big{[}\mathcal{O}_{\ell q}^{(3)}\big{]}_{prst} &=  \big{(}\overline{L_p}\,\gamma_\mu \tau^I \,L_{r}\big{)} \big{(}\overline{Q_{s}} \,\gamma^\mu \tau^I \,{Q_t}\big{)}\,,\quad\qquad
\big{[}\mathcal{O}_{\ell e qu}^{(1)}\big{]}_{prst} = \big{(}\overline{L^a_p}\, e_{rR}\big{)} \varepsilon_{ab} \big{(}\overline{Q^b_s} \,  u_{tR}\big{)}\,,\\[0.3em]
\big{[}\mathcal{O}_{\ell e dq}\big{]}_{prst} &=  \big{(}\overline{L^{a}_p} \,e_{rR}\big{)} \big{(}\overline{d_{sR}}  \,{Q_t}^a\big{)}\,,\quad\qquad\hspace*{2.9em}
\big{[}\mathcal{O}_{\ell e qu}^{(3)}\big{]}_{prst} = \big{(}\overline{L_p^a}\, \sigma_{\mu\nu}\,e_{rR}\big{)} \varepsilon_{ab} \big{(}\overline{Q^b_s}\, \sigma^{\mu\nu} \,u_{Rt}\big{)}\,,
\end{split}\end{aligned}$$ where $a,b$ are $SU(2)_L$ indices, $\varepsilon_{12}=-\varepsilon_{21}=1$, and $p,r,s,t$ are flavor indices. The matching at $\mu=m_b$ of this basis of operators onto the $SU(3)_c \times U(1)_\mathrm{em}$ invariant one, given in Eq. , tell us that $g_{V_R}^\ell$ cannot break LFU at dimension-6. This coefficient can only be generated by the LFU conserving operator $\mathcal{O}_{H u}= \big{(}H^\dagger i \overleftrightarrow{D}_\mu H\big{)}\big{(}\bar{u}_{pR} \gamma^\mu u_{rR}\big{)}$, being irrelevant for the following discussion [@Fajfer:2012jt; @Aebischer:2015fzz; @Bernard:2006gy]. The other Wilson coefficients have a 