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I maybe wrong, but; I though the 500 AHR was in fact and improved 500 jeffrey. Using 500 jeff brass and start loads.
But, I could be 100% wrong. Or 50% right I did google up some images to verify the rebated rim.

The 550 CZ is a great rifle. If you tire of the AHR, it could be rechambered for the 500 A Sguare.
I am pretty darn sure on that, but; this is all from memory. There is or was a short version of
the 500 A square thst should also work too. I mean as a backup plan.

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The .500 AHR case body is in fact within a couple of thousandths dimensionally of the .500 Jeffery, the two main differences are the .500AHR rim diameter was increased to the size of the .416 Rigby (still slightly rebated) and the case neck is .15” longer than the .500 Jeffery resulting in a .15” longer case length overall.
The case capacity are within 4 grains of water, 147.8 grains for the AHR and 143.7 grains for the Jeffery. Loads for one work in the other.
Call the .500 AHR “improved”, call it “updated”, call it “modernized”, the only difference that matters is a slightly longer neck and a rim that is less rebated.

Not sure why I would bother with the A Square......the .500 AHR or Jeffery with a 535 grain bullet at 2400’/sec in an 11.5lb rifle with no muzzle break is quite enough for me. (Yep, I’m a wimp... )
- Mike