  The fully self-consistent GW approximation is an established method for
electronic structure calculations. Its most serious deficiency is known to be
an incorrect prediction of the dielectric response. In this work we examine the
GW approximation for the homogeneous electron gas and find that problems with
the dielectric response are solved by enforcing the particle-number
conservation law in the polarization function. Previously reported data for the
ground-state energy were plainly contradicting each other well outside of
reported error bounds. Some of these results created a false impression of how
accurate the fully self-consistent GW approximation is. We resolve this
controversy by confirming that only Ref. [15] was reporting correct energy
data, and present values for other key Fermi-liquid properties.
