I finally snagged a Conn 5 mouthpiece and dissected it right quick. I see why no one liked it. It's awful.

The Conn 5 Experience

Rim shape = Flat as heck
Cup diameter ~ 25.3mm
Cup shape = Parabolic V-cup
Cup depth ~ Around the same as Bach 5G or JK Euphonium B-cup.
Throat ~ 5.8mm. I can't measure it directly, but it's below 6mm. Seriously.
Backbore is tiny. ~9.1mm at the opening. Garbage.
Total length = 84mm

THE SHANK!!!

11.8mm @ 1:20

No proprietary taper. Nothing special. It took 50+ years for someone to measure a standard taper 11.8mm shank. Good job, Earth. It's similar to but not really compatible with "Euro shank". Besson medium shank is approximately 11.7mm @ 1:19, so the difference is a little over 2mm of insertion depth and a slightly mismatching taper.

Any reputable mouthpiece guy can make you a Euphonium mouthpiece with an 11.8mm shank. You could specify that you want a small tenor shank backbore, but uhh...why? Here's your chance to beat me to having the first ever custom mouthpiece made for a Connstellation Euphonium. I'm waiting until I've got the Artist era Baritone shank reverse engineered since I don't own a 24-I. Yet.