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The repulsive three-body force between the lambda ($\Lambda$) hyperon andmedium nucleons is a key element in solving the hyperon puzzle in neutronstars. We investigate the binding energies of $\Lambda$ hyperon in hypernucleito verify the repulsive $\Lambda$ potentials from the chiral effective fieldtheory ($\chi$EFT) employing the Skyrme Hartree-Fock method. We find that the$\chi$EFT $\Lambda$ potential with the $\Lambda NN$ three-body forcesreproduces the existing hypernuclear binding energy data, whereas the $\Lambda$binding energies are overestimated without the $\Lambda NN$ three-body force.Additionally, we search for the parameter space of the $\Lambda$ potentials byvarying the Taylor coefficients of the $\Lambda$ potential and the effectivemass of $\Lambda$ at the saturation density. Our analysis demonstrates that theparameter region consistent with the $\Lambda$ binding energy data spans a widerange of the parameter space, including even more repulsive potentials than the$\chi$EFT prediction. We confirm that these strong repulsive $\Lambda$potentials suppress the presence of $\Lambda$ in the neutron star matter. Wefound that the $\Lambda$ potentials repulsive at high densities are favoredwhen the depth of the $\Lambda$ potential at the saturation density,$U_\Lambda(\rho_0)=J_\Lambda$, is $J_\Lambda\gtrsim-29~\text{MeV}$, whileattractive ones are favored when $J_\Lambda \lesssim -31~\text{MeV}$. Thissuggests that the future high-resolution data of hypernuclei could rule out thescenario in which $\Lambda$s appear through the precise determination of$J_\Lambda$ within the accuracy of $1~\text{MeV}$.

Repulsive $\Lambda$ potentials in dense neutron star matter and binding energy of $\Lambda$ in hypernuclei