  This work is an attempt to model the $4n$ response function of a recent RIKEN
experimental study of the double charge exchange $^4$He($^8$He,$^8$Be)$^4$n
reaction in order to put in evidence an eventual enhancement mechanism of the
zero energy cross section, including a near-threshold resonance. This resonance
can indeed be reproduced only by adding to the standard nuclear Hamiltonian an
unphysically large T=3/2 attractive 3n-force which destroys the neighboring
nuclear chart. No other mechanisms like cusps or related structures were found.
