2001 tribute brake whine

For about a year, we've had a brake whine that goes away when the brake
pedal is touched. This whine is getting worse. Now often touching the
brake pedal doesn't stop the whine. It takes a few seconds to go away
after the engine is turned off. The brakes work well. Over a period of
maybe 20-30 seconds of driving, the whine will slowly peter out. But
when I use the brakes, and then release them (i.e. slowing down and then
speeding up), the whine returns.

A service garage had another report of this problem. They said they
replaced everything and the whine was still there. Of course I doubt
that they replaced EVERYTHING. That's what they said.

I'm stumped.