No it won't work. NFC chips are powered with inductance. If you fold the induction coil that powers the chip the power supply will break so the chip will be useless. You could just make your own inductor but it i a horribly tough project to make especially if you are a beginner (including myself). So bottom line if you can't make another induction coil don't fold the existing. Also if you are making the ring out of metal you must think about the magnetic properties of the material. Here's a link that will help you
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