Man… this is not looking good… that sucks. I really hope it’s not a problem with the T5577 chip, but it is not looking good.
I would try doing an lf tune command on your proxmark3… with the iceman firmware this will start a looping voltage measurement that should look something like this;
[usb] pm3 --> lf tune
[=] Measuring LF antenna at 125.00 kHz, click pm3 button or press Enter to exit
[=] 30499 mV / 30 V / 31 Vmax
As you bring your NExT to the ProxLF antenna in a parallel orientation to the barrel coil, you should see the voltage drop… sometimes by only a couple hundred mV only… that’s what my xEM does when I bring it close… goes from 30500 (average) to 30200 in the “ideal” position. If you don’t see much change at all then it might be a chip to antenna connection problem, which would mean game over and we send you a replacement with our sincere appologies.
If you do see some change, then I would try to write a new EM410x ID to the T5577 with the Proxmark using the clone command;
lf em 410x clone --id 0F0368568B
Try to write it several times ensuring position of the LF antenna. With the ProxLF you will want to keep the NExT parallel to the coil barrel of course. After this, you should be able to issue lf search and get back an EM type ID. If that fails, try lf t5 detect and see if it detects anything at all.