Ha! Some time ago I read an article where they proposed an experiment where they wanted to test exactly what you're asking. Here is the article and here is the paper.
The idea is the following:
Place a shielded neutron detector near a neutron reactor and place a thick wall between them. This wall should be thick enough so that they normally don't reach the detector. In the reactor, the neutrons have a certain probability to transition into a "sterile" state where they exist in another brane. In this brane they are not affected by the wall.
Then, place the detector in different distances from the neutron source. When you see a $1/r^2$ law then this is because the shielded detector detects neutrons that leak from the other brane into ours. They also claim that if the neutrons leak from another brane to ours then you can additionally measure both of the following effects:
- the detected neutrons flux should not vary if they add extra shielding to the detector and
- since the coupling between the branes and therefore the rate at which neutrons transition into our brane depends on gravitation on earth, the detected neutron flux should depend on the earth's position on the orbit around the sun.
Referring to the picture below the trajectory of the neutron is then our brane (reactor) $\to$ hidden brane $\to$ our brane (detector).
To do this experiment, they plan to build an additional small $(36cm^3)$ neutron detector near an already existing neutron reactor - the neutron reactor of the institut Laue Langevin in Grenoble, France.
However, this is only a proposed experiment, I don't know if they will build it soon.