Hey there,
Even if this were possible (ultimately I believe it's going to come down to a race condition - will FS clean up the old channel's UUID and handles before starting the next one), it seems like a very bad idea. FS bases all channel actions on the premise of having unique session uuids. If you share UUIDs between channels, you're only going to cause yourself more headache. Assuming you're wanting to be able to identify that these three physical calls are all part of the same logical call, you'll be much better off generating a unique ID in your application somewhere and setting it on all legs as a channel var. Then you can simply use that identifier in your application, if you want this behavior.
As to the direct answer to your question, though - as mentioned above I believe it'll be a race condition at best, and a consistent failure at worst (and I'm quite confident the FS guys are going to tell you not to do this).
Best,
Colin