I'm working on the search for the EDM of the muon. Essentially it's much harder to search for the proton EDM than the neutron EDM. All EDM searches rely on a strong electric field applied to the particles. Because neutrons are neutral they are easily stored in some volume for a long time. You cannot so easily store protons because the moment you apply some E-field you start accelerating them. That's why you need to build a large storage ring with magnetic/electric focusing and so on. This brings numerous challenges that you don't have for the neutron. This, combined with the fact that we don't expect much different novel physics for the proton that won't be seen in the neutron has led to the focus on the neutron EDM, while the proton was left behind.

The usual quote is that for the proton we can reach sensitivities up to 10^-29 (around three orders of magnitude lower than the current nEDM limit), but thats only the statistical sensitivity. The systematic effects that would spoil that come much earlier and this limit is close to science fiction at this point. For example, if you have a magnetic field in the order of attotesla in the region of the storage ring it will dominate the measurement.

Would be happy to answer more edm questions :)