  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.
