  A hybrid model that employs the hadron-string transport model UrQMD and the
(3+1)D relativistic viscous hydrodynamic code vHLLE, is used to investigate the
beam energy dependence of the correlation coefficient $\rho(v^{2}_{2},[p_{T}])$
between the average transverse momentum $[p_{T}]$ of hadrons emitted in an
event and the square of the anisotropic flow coefficient $v_2^2$. For Au+Au
collisions, the model predicts characteristic patterns for the energy and
event-shape dependence of the variances for $[p_{T}]$ and $v_n^2$ (${\rm
Var}([p_T])$ and ${\rm Var}(v_2^2)$), and the covariance of $v_n^2$ and
$[p_{T}]$ (${\rm cov}(v_{2}^{2},[p_T])$), consistent with the attenuation
effects of the specific shear viscosity $\eta/s$. In contrast,
$\rho(v^{2}_{2},[p_{T}])$ is predicted to be insensitive to the beam energy but
sensitive to the initial-state geometry of the collisions. These observations
suggest that a precise set of measurements for ${\rm Var}([p_T])$, ${\rm
Var}(v_2^2)$, ${\rm cov}(v_{2}^{2},[p_T])$ and $\rho(v^{2}_{2},[p_{T}])$ as a
function of beam-energy and event-shape, could aid precision extraction of the
temperature and baryon chemical-potential dependence of $\eta/s$ from the
wealth of Au+Au data obtained in the RHIC beam energy scan.
