  It has been shown recently that the anomalies observed in $\bar B\to
D^{(\ast)}\tau\bar\nu_\tau$ and $\bar B\to \bar K\ell^+\ell^-$ decays could be
resolved with just one scalar leptoquark. Fitting to the current data on
$R(D^{(\ast)})$ along with acceptable $q^2$ distributions in $\bar B\to
D^{(\ast)}\tau\bar\nu_\tau$ decays, four best-fit solutions for the operator
coefficients have been found. In this paper, we explore the possibilities of
how to discriminate these four solutions. Firstly, we find that two of them are
already excluded by the decay $B_c^-\to \tau^-\bar\nu_\tau$, because the
predicted decay widths have already overshot the total width $\Gamma_{B_c}$. It
is then found that the remaining two solutions result in two effective
Hamiltonians governing $b\to c \tau \bar\nu_{\tau}$ transition, which differ by
a sign and enhance the absolute value of the coefficient of $\bar c_L\gamma_\mu
b_L\,\bar \tau_L\gamma^\mu{\nu_\tau}_L$ operator by about $12\%$. However, they
give nearly the same predictions as in the SM for the $D^\ast$ and $\tau$
longitudinal polarizations as well as the lepton forward-backward asymmetries
in $\bar B\to D^{(\ast)}\tau\bar\nu_\tau$ decays. For the other observables
like $\mathcal B(B_c^-\to \tau^-\bar\nu_\tau)$, $\mathcal B(B_c^- \to
\gamma\tau^-\bar\nu_\tau)$, $R_{D^{(\ast)}}(q^2)$, ${\rm d}\mathcal B(\bar B\to
D^{(\ast)}\tau\bar\nu_\tau)/{\rm d}q^2$ and $\mathcal B(\bar B\to
X_c\tau\bar\nu_\tau)$, on the other hand, the two solutions give sizable
enhancements relative to the SM predictions. With measurement of $B_c^-\to
\tau^-\bar\nu_\tau$ at LHCb and refined measurements of observables in $\bar
B\to D^{(\ast)}\tau\bar\nu_\tau$ at both LHCb and Belle-II, such a specific NP
scenario could be further deciphered.
