Unlike a motor vehicle operators permit (driver's license) for which the permission to drive a car ceases at the end of the date shown on the card, the permission to permanently reside in the US does not have an expiration date (hence the name "permanent"). Permanent residency can be terminated in certain conditions but length of residency or an arbitrary date shown on a plastic card is not one of them.
So, no, there is no 6-month remaining validity requirement. Such a requirement is moot, meaningless, and nonsensical, even, since residency is ostensibly "permanent."
As with an expired passport, even if the permanent residency card itself is expired, CBP will still allow the passenger to re-enter the country assuming s/he is otherwise still admissible (ie. hasnt otherwise lost the right to permanent residency). In such situations expect to be sent to secondary inspection for a longer wait while they verify admissibility.
This is much more common situation than one might think and various US Embassy webpages have information on this. Here is one relatively recently dated such page:
What to do if your LPR card or ‘green card’ is Expired, Lost or Stolen? - U.S. Embassy in the Dominican Republic (usembassy.gov)
Will be useful for you to bring this document to check-in with you:
Reminder- LPR Boarding 20210305.pdf (cbp.gov)
You will be fine, but maybe build in extra time at the airport in case the airline check-in agents are not sure what to do about you and a supervisor or even a manager is needed to do research on Timatic once they have seen the CBP document and US Embassy webpage you show them.