  The nature of the ground state of the spin $S=1/2$ Heisenberg antiferromagnet
on the kagome lattice with breathing anisotropy (i.e., with different
superexchange couplings $J_{\vartriangle}$ and $J_{\triangledown}$ within
elementary up- and down-pointing triangles) is investigated within the
framework of Gutzwiller projected fermionic wave functions and Monte Carlo
methods. We analyze the stability of the U(1) Dirac spin liquid with respect to
the presence of fermionic pairing that leads to a gapped $\mathbb{Z}_{2}$ spin
liquid. For several values of the ratio $J_{\triangledown}/J_{\vartriangle}$,
the size scaling of the energy gain due to the pairing fields and the
variational parameters are reported. Our results show that the energy gain of
the gapped spin liquid with respect to the gapless state either vanishes for
large enough system size or scales to zero in the thermodynamic limit.
Similarly, the optimized pairing amplitudes (responsible for opening the spin
gap) are shown to vanish in the thermodynamic limit. Our outcome is
corroborated by the application of one and two Lanczos steps to the gapless and
gapped wave functions, for which no energy gain of the gapped state is detected
when improving the quality of the variational states. Finally, we discuss the
competition with the "simplex" $\mathbb{Z}_{2}$ resonating-valence-bond spin
liquid, valence-bond crystal, and nematic states in the strongly anisotropic
regime, i.e., $J_{\triangledown} \ll J_{\vartriangle}$.
