OK, looked back up in the thread.
Twist rate is 1/54"
You note that the bore is .673/4", and the grove is .695", the front of the chamber before the 'forcing cone' or throat is .765"

The difference between the mouth of that chamber (Can you include please the distance from breech face to the line where you measured the .695" diameter?) and your groove diameter looks like .070"
Half that is .035"
Paper case walls (from my Cheddite 16-bore paper cases) are about .025" thick - and I've seen plastic cases that measured about the same as my paper cases.

Looks like a paper-case 14-bore for sure.

If you're going to be running those paradox-style conicals, lube the hell out of them.
Judging from the looks of that rifling and the conveniently modern-looking "forcing cone" (I'd bet you a carne asada burrito and a large horchata, that chamber has been re-cut some time in the last 50 years), Daryl's suggestion of running patched roundball would likely work out wonderfully in this rifle.
I'd try that right away!

Cheers
Tinker