If they had one giant ten-armed coat that all five of them were wearing simultaneously, and they took it off, how would one say "they take off their coat"? Would it be the same ("leur manteau") or is there a different way to express that distinction?
I understand that I've given a pretty absurd hypothetical, but the principle certainly could apply to more realistic scenarios.
In spoken French, I don't believe there is an aural difference between leur manteau and leurs manteaux - they sound exactly the same.
However, is it not possible that the singular version means each person removes one coat and one coat only, while the plural version can mean one coat, or two or more coats, which is quite possible in very cold, wet weather: A rain coat, an over-coat (trench-coat or otherwise), and then a dinner jacket, so at the end the person is wearing just a shirt.
In cold weather, I wear layers of clothing, with none very heavy, which enables controlling body temperature with more adaptability, since I can put on or take off just enough garments to be very comfortable.
Since we can't tell how many coats each of them is wearing, it should accept both transcriptions.
Admittedly it would be odd for multiple people to each be wearing multiple coats, but not as odd as what Duolingo sometimes has ducks and elephants do. Reported 19 June 2018.
In French, the singular is used to say that they have one coat each. The English sentence doesn't specify that it's one each, so both "leur manteau" and "leurs manteaux" are accepted, but it seems likely (and we want to teach the concept), so "leur manteau" is the "Best" answer.