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Helioseismology has revealed that the rotation rate of the solar convection zone decreases in a layer of strong radial shear just below the photosphere, known as the near-surface shear layer (NSSL). One puzzling aspect of this region is the Sun’s photospheric emissivity, which is observed to be nearly uniform in latitude, in contrast to predictions from theory and numerical simulations of rotating convection. In this work, we investigate how rotating convection responds to the imposition of a latitudinally varying heat flux at the base of a convective layer. The NSSL is thought to represent a transition from a buoyancy-dominated regime near the surface to a rotation-dominated regime at greater depth. Motivated by this, we conduct a suite of spherical, three-dimensional nonlinear simulations of rotating convection spanning both high-Rossby-number (buoyancy-dominated) and low-Rossby-number (rotation-dominated) regimes. At the base of each model convection zone, we impose a heat flux whose latitudinal variation opposes the pattern that the system would naturally establish, warm poles and cool equator. We identify a threshold Rossby number above which convective motions efficiently mix heat laterally, erasing the imposed latitudinal flux variation from imprinting at the surface. The resulting differential rotation weakens and the thermal wind balance is maintained, but breaks down at sufficiently high Rossby number, where inertial effects play a role. We interpret these results in the context of the NSSL, which exhibit weakened differential rotation relative to the deep convection and little to no latitudinal variation in photospheric emissivity.