It sounds like the printer has a firmware defect that HP cannot resolve. Normally when a fax machine does not answer messages that it normally would that means there is a bad message stuck in the queue. A NVRAM/COLD RESET would clean up an issue like that. You mentioned that your printer can continue recieving once you manually wake it up so I wouldnt necessarily think a bad fax message is to blame.
Considering your printer is working fine for everything but fax now you might want to consider migrating away from physical fax lines and checking out Online Fax. An online fax vendor will allow you to send and recieve everything via email meanwhile they will handle the physical fax message sending and recieving. Its a lot easier and more stable in the faxing world we live in today.
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