2 3/8" is the standard these days but some of these posts are 75+ years old (old manufacturing facility) and I guess the standard pole size back then was different than it is today.
In that case, I'd say you could maybe go with a 2" square post cap. It should still work, you just may need to drill and tap new set screw locations that are closer to center or use longer set screws.
Another option might be to find some galvanized 2.25" OD pipe with a 1/8" wall, cut them to whatever length you need, slit them, and use them as adaptors between the posts and post brackets.