  A hypothesis of absolutely stable strange hadronic matter composed of
$\Lambda(1405)$ baryons, here denoted $\Lambda^*$, is tested within many-body
calculations performed using the Relativistic Mean-Field approach. In our
calculations, we employed the $\Lambda^*\Lambda^*$ interaction compatible with
the $\Lambda^*\Lambda^*$ binding energy $B_{\Lambda^*\Lambda^*}=40$~MeV given
by the phenomenological energy-independent $\bar{K}N$ interaction model by
Yamazaki and Akaishi (YA). We found that the binding energy per $\Lambda^*$, as
well as the central density in $\Lambda^*$ many-body systems saturates for mass
number $A\geq120$, leaving $\Lambda^*$ aggregates highly unstable against
strong interaction decay. Moreover, we confronted the YA interaction model with
kaonic atom data and found that it fails to reproduce the $K^-$ single-nucleon
absorption fractions at rest from bubble chamber experiments.
