Never has this model in my hands (sadly so). However it was my understanding that this model had a receiver like a tenor trombone, say King 3B or Bach 36. Hence this photo confuses me:
Looks like there is an adapter. What size of mouthpiece would that be for?
The Conn 24I has a "Remington shank" receiver like older Conn 88H trombones. This is larger than tenor shank and smaller than bass trombone. It is similar to the "euro-shank" on older Bessons and American Willson 2900's, but it is a slightly different pitch. The adapter brings the Remington shank back to nornmal tenor-shank. Unless you are going to use an old Remington shank mouthpiece from Conn (I have two for my 24I) use the adapter and a tenor shank mouthpiece.
By the way, I have played 24I's off and on since 1965 and own a fine example again. These horns can have a very nice sopund, but they do not respond well to modern large bore mouthpieces like the Wick 4AY, 4BS, or 5BS. If you choose to use a modern mouthpiece, think of a Bach 6 1/2AL (my fav) or perhaps a 5G.
Interesting news to me. I have a Conn Remington large stem trombone mouthpiece given to me by Søren. It makes my large Jupiter tenor sound way better and more alive than does the almost similar DW4AL.
I haven’t ever been able to make anything sound well on Bach 6 1/2 A or DW 6B. However I have had the experience of a larger mouthpiece killing the response, which surprised me very much. My G bass trombones and my circular Eb bass valve trombone (named a fanfare helicon) by Kühnl & Hoyer become very slow when using the DW SM2AM. A DW4BS works better.