  In many theoretical frameworks our visible world is a $3$-brane, embedded in
a multidimensional bulk, possibly coexisting with hidden braneworlds. Some
works have also shown that matter swapping between braneworlds can occur. Here
we report the results of an experiment - at the Institut Laue-Langevin
(Grenoble, France) - designed to detect thermal neutron swapping to and from
another braneworld, thus constraining the probability $p^2$ of such an event.
The limit, $p<4.6\times 10^{-10}$ at $95 \%$ C.L., is $4$ orders of magnitude
better than the previous bound based on the disappearance of stored ultracold
neutrons. In the simplest braneworld scenario, for two parallel Planck-scale
branes separated by a distance $d$, we conclude that $d>87$ in Planck length
units.
