  Thanks to the recent careful revisit of the theoretical prediction of the
$B_c$ meson lifetime, the conservative upper bound on the branching ratio (BR)
of $\tau \nu$ mode is found to be $\simeq 63\%$ due to the large charm quark
mass uncertainty. Although it is well known that a charged Higgs ($H^-$)
interpretation of the $R_{D^{(*)}}$ anomaly is excluded by the previously
proposed bounds, BR$(B_c\to\tau\nu)\le 30\%$ and $\le10\%$, $H^-$ can still
explain the anomaly within $1\sigma$ if we adopt the 63$\%$ one. The scalar
contribution is also favored by the polarization data $F_L^{D^*}$ measured at
the Belle. Since the implied NP scale is within the reach of the Large Hadron
Collider (LHC), collider searches are powerful tools to test the scenario. For
instance, the $\tau\nu$ resonance search has already put the more stringent
bound for $m_{H^-}\ge 400\,$GeV. In this work we revisit the further lighter
mass range, $180\,$GeV$\le m_{H^-}\le 400\,$GeV which has not been covered yet.
We will see that a combination of the conventional stau search and low mass
flavor inclusive and bottom flavored di-jet resonance searches can place a new
limit on the interpretation. We summarize the current status of the low mass
region and discuss the future sensitivity in the high luminosity (HL)-LHC based
on the existent collider constraints.
