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I've got my hands on a 1957 Besson Brevete tuba in BBb... one of the goofy-looking ones with a recording bell. It's got the nice, fat sound of a BAT without the size of a BAT (except the 24" bell) and I love it.

I've noticed I have to pull the main slide out pretty far to be in tune... only about an inch and a half of both legs is still in the outer tubes. That being said, the tuning slide moves freely. A little too freely. When I pull out to be in tune and start to play, it will slowly shimmy its way down and fall out with a clang. It takes maybe 5 minutes of spirited playing for this to happen. The slide hasn't suffered any dents yet because I've managed to catch it before it hit the floor, but it's getting annoying. I've tried 3 different slide greases to no avail.

How does one remedy this problem?

You could probably have it expanded a little bit by a good repair tech. As for the tuning, I wonder if it's high pitch? You can have the slide extended to bring it down to pitch.