Which sentence is grammatically correct?
how long the couple remain faithful is something questionable.
how long the couple remains faithful is something questionable.
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"Couple" is a collective noun and can be treated as either singular or plural. So that means both of your examples are correct. Here's another example from Cambridge Dictionaries Online:
An elderly couple live (US lives) next door.
The parenthetical supplement "US lives" implies that the singular agreement is more common in the US than in the UK.