Great question — answering as the UphillTech team:
You could try using impact to pop the handles off, as diagrammed below. Loop some nylon line around the three handles. Put a steel shaft, such as a screwdriver, through the loops. Leave the cords slack, and then give a sudden tug, using a tool such as a hammer's claw (or a slide hammer , if you have or care to fabricate one), to yank on the lines. Of course, this requires judgement -- too gentle a tug does nothing, and too hard might break the faucet. Start off gently , and carefully increase the strength of each tug.
Full answer by DrMoishe Pippik: https://diy.stackexchange.com/a/320554 (CC BY-SA 4.0).
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