if it was a wastegate rattle, it'd be less likely to disappear after just a few seconds. unless you have a vacuum problem. no idea why they decided replacing the turbo is the best course of action. then again, trying to diagnose an engine over the internet is always tricky. but seeing how the turbo swap didn't fix your issue...

i have no advice on your lease. but if you feel you're getting yourself into trouble by buying the car, then maybe don't. all i can say is that i wouldn't trust either of the dealerships you visited with actually keeping your car in excellent working order. i'd expect a little more than just a bunch of wild and somewhat misguided stabs in the dark from an authorised service agent. especially if they quickly jump to conclusions that lead to replacements of expensive engine parts.

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