Mystery pipe dripping into laundry tub

[Maugham]

My washing machine and laundry tub are in the basement. The cold water supply comes down from the joists. About halfway down there is a weird T joint which runs an extra pipe into the laundry tub. The extra pipe is open and seems to have some kind of weird shut off valve but drips constantly into the tub. I've had 2 plumbers here and neither could figure out what it was for for and what the best way would be to stop the constant dripping. One tried to tell me that I needed an expansion tank (whatever that is). I'm pretty sure that the simplest and best solution would be to replace the T at the top with an elbow joint and then replace the existing frankenpipe with a simple run down to the shut off valve for the washing machine.

 

[Gasmk1]

Can’t be certain from photos but looks like some sort of safety relief valve