  The Generalized Gibbs Ensemble (GGE) is relevant to understand the
thermalization of quantum systems with an infinite set of conserved charges. In
this work, we analyze the GGE partition function of 2D Conformal Field Theories
(CFTs) with a U(1) charge and quantum Benjamin-Ono$_{2}$ (qBO$_{2}$) hierarchy
charges. We use the Alday-Gaiotto-Tachikawa (AGT) correspondence to express the
thermal trace in terms of the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of
descendants, which diagonalizes all charges. We analyze the GGE partition
function in the thermodynamic semiclassical limit, including the first order
quantum correction. We find that the equality between GGE averages and primary
eigenvalues of the qBO$_{2}$ charges is attainable in the strict large $c$
limit and potentially violated at the subleading $1/c$ order. We also obtain
the finite $c$ partition function when only the first non-trivial charge is
turned on, expressed in terms of partial theta functions. Our results should be
relevant to the eigenstate thermalization hypothesis for charged CFTs, Warped
CFTs and effective field theory descriptions of condensed matter systems.
