Here's some more info on the 444 rimless. Looks like the 44??? is just a short 444 rimless.
A "Rimless" 444 Marlin... - Page 2 - Shooters Forum
Don't know what's up with the link. Go to the box at the top of the page and click DAW A "Rimless" 444 Marlin.
I gave up on finding a Handi 44 mag barrel, so a 44 mag barrel is on the way. I'll make it into another insert for testing.