  We theoretically assess the conjecture proposed by Kovtun, Son, and
Starinets, stating that the ratio $\eta/s$ of the shear viscosity $\eta$ to the
entropy density $s$ has the lower bound as $\eta/s\ge\hbar/(4\pi
k_{\mathrm{B}})$. In the normal state of a mass-imbalanced ultracold Fermi gas,
consistently including strong-coupling corrections to both $\eta$ and $s$
within the self-consistent $T$-matrix approximation, we evaluate $\eta/s$ over
the entire BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation)
crossover region, in the presence of mass imbalance. We find that $\eta/s$
achieves the minimum value $4.5\times \hbar/(4\pi k_{\mathrm{B}})$, not at the
unitarity, but slightly in the BEC regime, $(k_{\mathrm{F}}a_s)^{-1}\simeq
0.4>0$ (where $a_s$ is the $s$-wave scattering length, and $k_{\mathrm{F}}$ is
the Fermi momentum). In contract to the previous expectation, we find that this
lower bound is almost independent of mass imbalance: Our results predict that
all the mass-balanced $^6$Li-$^6$Li and $^{40}$K-$^{40}$K mixtures and the
mass-imbalanced $^{40}$K-$^{161}$Dy mixture give almost the same lower bound of
$\eta/s$. We also point out that the two quantum phenomena, Pauli blocking and
bound-state formation, are crucial keys for the lower bound of $\eta/s$.
