  The question of the competition between $\Lambda$ and $\Xi^{0,-}$ in the
ground-state of multi-strange hypernuclei is addressed within a
non-relativistic density functional approach, partially constrained by
ab-initio calculations and experimental data. The exploration of the nuclear
chart for $10<Z<120$ as a function of the strangeness number is performed by
adding hyperons to a nuclear core imposing either conserved total charge $Q$ or
conserved proton number $Z$. We find that almost all $\Lambda$ hypernuclei
present an instability with respect to the strong interaction decay of
$\Lambda$ towards $\Xi^{0,-}$ and that most of the instabilities generates
$\Xi^-$ (resp. $\Xi^0$) in the case of conserved total charge $Q$ (resp. proton
number $Z$). The strangeness number at which the first $\Xi^{0,-}$ appear is
generally lower for configurations explored in the case of conserved $Q$
compared to the case of conserved $Z$, and corresponds to the crossing between
the $\Lambda$ and the neutron or proton chemical potentials. About two to three
hundred thousands pure $\Lambda$ hypernuclei may exist before the onset of
$\Xi^{0,-}$. The largest uncertainty comes from the unknown $\Lambda\Xi$
interaction, since the $N\Lambda$ and the $N\Xi$ ones can be constrained by a
few experimental data. The uncertainty on the $\Lambda\Xi$ interaction can
still modify the previous estimation by 30-40\%, while the impact of the
unknown $\Xi\Xi$ interaction is very weak.
