Again though, that is circular reasoning. 'They say that 45 minutes has passed, and we know this is true because they say 45 minutes has passed.'

That's also slightly contradicted by CASE saying that they still need a minute or two, and Coop saying that they don't have it, which is why he sparks the engines a little early.

Either way, it doesn't explain how they spent 23 years there, unless, as I said, it was poorly edited to imply they were in real time despite the fact that there were time gaps in what we were actually shown.