I still think that barrel is butt reaming ugly

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I've heard that it is a "sleeve" that's been placed over the barrel

I dunno?
I don't think that you would be able to take a standard heavy 16" barrel and make it exactly with the profile you are looking for. I haven't physically handled one of them, but it looks like at the point marked #3 in the pic, that it is even larger diameter than a HB profile?
If not, just use the pic and a lathe and eliminate anything on your barrel that doesn't look like the pic

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I think after all the work/money you would still end up with a barrel too ugly to put on a rifle.