  The correlation between the event mean-transverse momentum
$[p_{\mathrm{T}}]$, and the anisotropic flow magnitude $v_n$,
$\rho(v^{2}_{n},[p_{T}])$, has been argued to be sensitive to the initial
conditions in heavy-ion collisions. We use simulated events generated with the
AMPT and EPOS models for Au+Au at $\sqrt{\textit{s}_{NN}}$ = 200 GeV, to
investigate the model dependence and the response and sensitivity of the
$\rho(v^{2}_{2},[p_{T}])$ correlator to collision-system size and shape, and
the viscosity of the matter produced in the collisions. We find good
qualitative agreement between the correlators for the string melting version of
the AMPT model and the EPOS model. The model investigations for
shape-engineered events as well as events with different viscosity ($\eta/s$),
indicate that $\rho(v^{2}_{2},[p_{T}])$ is sensitive to the initial-state
geometry of the collision system but is insensitive to sizable changes in
$\eta/s$ for the medium produced in the collisions. These findings suggest that
precise differential measurements of $\rho(v^{2}_{2},[p_{T}])$ as a function of
system size, shape, and beam-energy could provide more stringent constraints to
discern between initial-state models and hence, more reliable extractions of
$\eta/s$.
