Probably to keep the main tuning slide away from your front side (shirt, gut, etc.) when holding it. Especially if the slide is pulled out some and has grease on the leg. There are a bunch of different type things on euphoniums that do just that, but each one looks a little different. You find these mainly on euphoniums which have a trigger to activate the main tuning slide to flatten notes that may be sharp.
Here is a picture of my horn after a few mods were done last year:

You can see what I have to cover the main tuning slide leg. It looks a little different from yours, I have a screw to loosen the covering device and move it up and down. Looks like you have some push pins to change it a certain amount. I guess this must be a "new" feature with Adams horns. Can anyone else verify that comment.
Roger - did you specifically ask for this feature?
John Morgan
The U.S. Army Band (Pershing's Own) 1971-1976
Adams E3 Custom Series Euphonium, Wessex EP-100 Dolce Euphonium, 1956 B&H Imperial Euphonium
Adams TB1 Tenor Trombone, Yamaha YBL-822G Bass Trombone
Kingdom of the Sun (KOS) Concert Band, Ocala, FL (Euphonium)
KOS Brass Quintet (Trombone, Euphonium)
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