I cannot find the citation right now, but the p and n EDMs are expected to be close to each other ~1e-32 e.cm. One part is that they are uud and udd and the other thing to consider is that the quarks make up only ~2% of the proton/neutron mass. Most is binding energy and a soup of virtual quarks and gluons and in that regard they are even more similar I think. I am not very familiar how theoreticians calculate the EDM of such complex particles though.

On a side note, 'ultra cold neutrons' are a super interesting type of matter. Their energy is so low that they can be stored in bottles and are transported through tubes using turbines and mechanical valves.