  Two different kinetic theories [J. Solsvik and E. Manger (SM-theory), Phys. Fluids \textbf{33}, 043321 (2021) and V. Garz\'o, J. W. Dufty, and C. M. Hrenya (GDH-theory), Phys. Rev. E \textbf{76}, 031303 (2007)] are considered to determine the shear viscosity $\eta$ for a moderately dense granular binary mixture of smooth hard spheres. The mixture is subjected to a simple shear flow and heated by the action of an external driving force (Gaussian thermostat) that exactly compensates the energy dissipated in collisions. The set of Enskog kinetic equations is the starting point to obtain the dependence of $\eta$ on the control parameters of the mixture: solid fraction, concentration, mass and diameter ratios, and coefficients of normal restitution. While the expression of $\eta$ found in the SM-theory is based on the assumption of Maxwellian distributions for the velocity distribution functions of each species, the GDH-theory solves the Enskog equation by means of the Chapman--Enskog method to first order in the shear rate. To assess the accuracy of both kinetic theories, the Enskog equation is numerically solved by means of the direct simulation Monte Carlo (DSMC) method. The simulation is carried out for a mixture under simple shear flow, using the thermostat to control the cooling effects. Given that the SM-theory predicts a vanishing kinetic contribution to the shear viscosity, the comparison between theory and simulations is essentially made at the level of the collisional contribution $\eta_c$ to the shear viscosity. The results clearly show that the GDH-theory compares with simulations much better than the SM-theory over a wide range of values of the coefficients of restitution, the volume fraction, and the parameters of the mixture (masses, diameters, and concentration). 
Assessment of kinetic theories for moderately dense granular binary   mixtures: Shear viscosity coefficient