models have appeared to estimate the ground-state binding energies of double-$\Lambda$ species: Based on the old data of $_\Lambda^{ }$$_\Lambda^6$He and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be, Takaki [*et al.*]{} [@Takaki89] applied a simplified version of the $\alpha+x+ \Lambda + \Lambda$ cluster model to $A=6 - 10$ systems in which they put several angular momentum restriction and neglected rearrangement channels. Bodmer [*et al.*]{} [@Bodmer84; @Bodmer87] performed variational Monte Carlo calculations for $\alpha+ \Lambda+\Lambda$ and $\alpha+\alpha+ \Lambda+ \Lambda$ to investigate consistency between the $\Lambda \Lambda$-binding energies, $B_{\Lambda \Lambda}$($_\Lambda^{ }$$_\Lambda^6$He) and $B_{\Lambda \Lambda}$($_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be), although their old data should be now updated. In the latest stage of this work, we encountered with the Faddeev-Yakubovsky calculations of $_\Lambda^{ }$$_\Lambda^6$He and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be by Filikhin and Gal [@Gal01] who restricted the equations within the $s$-wave. They compared the results with our previous cluster-model calculation [@Hiyama97] which was performed with wider model space but the stronger $\Lambda \Lambda$ interaction strength. In our previous work [@Hiyama97], $\Lambda \Lambda$ binding energies have been calculated for $_\Lambda^{ }$$_\Lambda^6$He and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be in the framework of the $\alpha+\Lambda+\Lambda$ three-body model and the $\alpha+\alpha+ \Lambda+ \Lambda$ four-body model, respectively, where the adopted $\Lambda \Lambda$ interaction is taken to be considerably attractive on the basis 