Well, I'm an AMCIT, and I have mine... but:
I'm in Germany, covered under some very specific treaty agreements. I have a WBK, and I need the WBK to get the EFP. When I leave here, I have to turn in both, although I can take my Jagdschein home. Taking it home will make it easier to renew if I come back.
I fully admit I am in a special case. Contact me off line, and I can better explain.
Keep in mind: All the EFP really does is explain (in the appropriate language) to other member countries that your firearms are appropriately registered in a member country. Yes, when Brexit happened, the UK was no longer a member country, so you couldn't say your firearms, although they were listed in a national registry, were listed in a member registry.
My American firearms are listed in no national registry... and those handguns they made me register in California? It was tragic. Slipped right out of my hands in that boating incident off Avalon.
But, as above, the short answer is No.