  In a recent paper [PLB 837 (2023) 137669] we showed that all measured
($1s_\Lambda$, $1p_\Lambda$) pairs of $\Lambda$ binding energies in
$\Lambda$-hypernuclei across the periodic table, $12\leq A \leq 208$, can be
obtained from a $\Lambda$-nucleus optical potential with only two adjustable
$\Lambda N$ and $\Lambda NN$ parameters, associated with leading linear and
quadratic terms in the nuclear density, derived by fitting $^{16}_{~\Lambda}$N
binding energies. Here we extend the previous analysis by performing
least-squares fits to the full set of data points. Consequences of suppressing
$\Lambda NN$ interactions between `core' nucleons and `excess' neutrons are
studied and related predictions are made for ($1s_\Lambda$, $1p_\Lambda$)
binding energies in $^{40,48}_{~~~~\Lambda}$K, obtainable from upcoming
$^{40,48}$Ca($e,e'K^+$) JLab experiments. We find $\Lambda$-nucleus partial
potential depths of $D^{(2)}_\Lambda = -38.6\pm 0.8$~MeV ($\Lambda N$) and
$D^{(3)}_\Lambda = 11.3\pm 1.4$~MeV ($\Lambda NN$), with a total depth
$D_\Lambda = -27.3\pm 0.6$~MeV at nuclear-matter density
$\rho_0$=0.17~fm$^{-3}$, consistently with our previous results. Extrapolation
to higher nuclear densities and possible relevance to the `hyperon puzzle' in
neutron-star matter are discussed.
