$b\to c\tau\bar\nu_{e,\mu}$ contributions to $R(D^{(*)})$

The $R(D^{(*)})$ puzzle stands for a $\sim3\sigma$ violation of lepton flavor universality between the decay rates of $B\to D^{(*)}\tau\nu$ and $B\to D^{(*)}\ell\nu$, where $\ell=e,\mu$. If it is accounted for by new physics, there is no reason in general that the relevant neutrinos are, respectively, $\nu_\tau$ and $\nu_\ell$. We study whether the $\tau$ related rate could be enhanced by significant contributions to $B\to D^{(*)}\tau\nu_\ell$ from a class of operators in the Standard Model Effective Field Theory (SMEFT). We find the upper bounds from forbidden or rare meson decays imply that the contributions from the lepton flavor violating processes account for no more than about $4\%$ of the required shift. Yet, no fine-tuned flavor alignment is required for the new physics. Searching for the related high-$p_T$ process $pp\to\tau^\pm\mu^\mp$ can at present put a lower bound on the scale of the lepton flavor violating new physics that is a factor of $2.2$ weaker than the bound from meson decays. An exception to our conclusion arises from a specific combination of scalar and tensor SMEFT operators.