If the flight plan is not closed in due time - i.e. after the "endurance time" you wrote - then they basically have to trigger SAR, yes.
Jan that is true everywhere that I'm aware of, EXCEPT the one under discussion here! The UK does not have that system, and will not do S&R unless someone specifically requests it (known in the UK as a responsible person). Simply not closing a plan won't result in S&R in the UK.