  Density-balanced, widely separated quantum Hall bilayers at $\nu_T = 1$ can
be described as two copies of composite Fermi liquids (CFLs). The two CFLs have
interlayer weak-coupling BCS instabilities mediated by gauge fluctuations, the
resulting pairing symmetry of which depends on the CFL hypothesis used. If both
layers are described by the conventional Halperin-Lee-Read (HLR) theory-based
composite electron liquid (CEL), the dominant pairing instability is in the
$p+ip$ channel; whereas if one layer is described by CEL and the other by a
composite hole liquid (CHL, in the sense of anti-HLR), the dominant pairing
instability occurs in the $s$-wave channel. Using the Dirac composite fermion
(CF) picture, we show that these two pairing channels can be mapped onto each
other by particle-hole (PH) transformation. Furthermore, we derive the CHL
theory as the non-relativistic limit of the PH-transformed massive Dirac CF
theory. Finally, we prove that an effective topological field theory for the
paired CEL-CHL in the weak-coupling limit is equivalent to the exciton
condensate phase in the strong-coupling limit.
