  Multi-strange Ca, Sn and Pb hypernuclei with $\Lambda\Lambda$ pairing
interaction are investigated within the Hartree-Fock-Bogoliubov approach. The
unknown $\Lambda\Lambda$ pairing strength is calibrated to match with the
maximal value for the prediction of the $\Lambda$ pairing gap in uniform matter
for densities and isospin asymmetries equivalent to those existing in
multi-$\Lambda$ hypernuclei. In this way, we provide an upper bound for the
prediction of the $\Lambda$ pairing gap and its effects in hypernuclei. The
condensation energy is predicted to be about 3~MeV as a maximum value, yielding
small corrections on density distributions and shell structure. In addition,
conditions on both Fermi energies and orbital angular momenta are expected to
quench the nucleon-$\Lambda$ pairing for most of hypernuclei.
