attraction at intermediate distances. Its partial suppression by the Pauli principle acting on intermediate nucleon states in nuclear or neutron matter is responsible for the turnover from attraction to repulsion in the $\Lambda$-nuclear two-body force at a density well below $3\,\rho_0$ in nuclear matter. From fig.\[fig:ULambda1\] we also recall that the $\Lambda N$ attraction provided by NLO13 is slightly stronger than what is needed for the empirical hypernuclear shell-model potential. This sets the frame for constraining unknown pieces of the repulsive $YNN$ three-body force which in turn governs the behaviour of $\mu_\Lambda$ at high density. One might then ask whether alternative options exist which could imply different constraints and extrapolations for the 3BF sector. Such an alternative $YN$ potential (NLO19) has recently been studied [@Haidenbauer2019]. This potential is equivalent to NLO13 with respect to $\Lambda N$ and $\Sigma N$ scattering data, but it produces a stronger attractive $\Lambda$-nuclear potential than NLO13 at $\rho\simeq\rho_0$. This difference needs to be balanced by a more strongly repulsive three-body $\Lambda NN$ interaction. We have thus performed Brueckner calculations (for continuous choice) using NLO19, together with a 3BF readjusted to meet the hypernuclear $U_\Lambda(\rho=\rho_0) = -30$ MeV constraint. The effect at high density is to 