Abstract: Quark-model hyperon-nucleon and hyperon-hyperon interactions by the Kyoto-Niigata group are applied to the two-$\ensuremath{\Lambda}$ plus $\ensuremath{\alpha}$ system in a three-cluster Faddeev formalism using two-cluster resonating-group method kernels. The model fss2 gives a reasonable two-$\ensuremath{\Lambda}$ separation energy $\ensuremath{\Delta}{B}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}=1.41\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$, which is consistent with the recent empirical value, $\ensuremath{\Delta}{B}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{\mathrm{exp}}=1.01\ifmmode\pm\else\textpm\fi{}0.20\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$, deduced from the Nagara event. Some important effects that are not taken into account in the present calculation are discussed.