  We report on robust features of the longitudinal conductivity ($\sigma_{xx}$)
of the graphene zero-energy Landau level in presence of disorder and varying
magnetic fields. By mixing an Anderson disorder potential with a low density of
sublattice impurities, the transition from metallic to insulating states is
theoretically explored as a function of Landau-level splitting, using highly
efficient real-space methods to compute the Kubo conductivities (both
$\sigma_{xx}$ and Hall $\sigma_{xy}$). As long as valley-degeneracy is
maintained, the obtained critical conductivity $\sigma_{xx}\simeq 1.4 e^{2}/h$
is robust upon disorder increase (by almost one order of magnitude) and
magnetic fields ranging from about 2 to 200 Tesla. When the sublattice symmetry
is broken, $\sigma_{xx}$ eventually vanishes at the Dirac point owing to
localization effects, whereas the critical conductivities of pseudospin-split
states (dictating the width of a $\sigma_{xy}=0$ plateau) change to
$\sigma_{xx}\simeq e^{2}/h$, regardless of the splitting strength, superimposed
disorder, or magnetic strength. These findings point towards the non
dissipative nature of the quantum Hall effect in disordered graphene in
presence of Landau level splitting.
