  Many-brane universes are at the heart of several cosmological scenarios
related to physics beyond the Standard Model. It is then a major concern to
constrain these approaches. Two-brane Universes involving $SO(3,1)$-broken 5D
bulks are among the cosmological models of interest. They also allow
considering matter exchange between branes, a possible way to test these
scenarios. Neutron disappearance (reappearance) toward (from) the hidden brane
is currently tested with high-precision experiments to constrain the coupling
constant $g$ between the visible and hidden neutron sectors. When dealing with
the sub-GeV-scale quantum dynamics of fermions, any pair of braneworlds can be
described by a non-commutative two-sheeted space-time $M_4\times Z_2$ from
which $g$ emerges. Nevertheless, the calculation of the formal link between $g$
for a neutron and $SO(3,1)$-broken 5D bulks remains an open problem until now
although necessary to constrain these braneworld scenarios. Thanks to a
phenomenological model, we derive $g$ - for a neutron - between the two
braneworlds endowed with their own copy of the standard model in a
$SO(3,1)$-broken 5D bulk. Constraints on interbrane distance and brane energy
scale (or brane thickness) are discussed. While brane energy scale below the
GUT scale is excluded, energy scale up to the Planck limit allows neutron
swapping detection in forthcoming experiments.
