Forgive me if this sounds presumptuous, but is this experiment literally trying to answer "does antimatter attract normal matter"? To me, this seems like a pretty simple thing to test–why hasn't this been done before?
If you just generated individual relativistic anti-neutrons and tried to watch them follow ballistic trajectories in gravity, I suspect they'd just look like perfectly straight lines up to the limits of your detector resolution. Not sure though.
(The only other quasi-stable neutral particles are neutrinos, and those are essentially non-interacting with any detector. They leave relativistically and you'll never see them again.)
Also, you'll need the neutrinos to start out at rest; they tend to travel near the speed of light.
So a fun problem that comes up is how long will it take for the neutrons to collide assuming they're only being accelerated by the mass of the container.