You should have closed the case after the minimum FOUR days.

The onus is on the buyer to prove it's a fake. Which requires getting a third party, competent person to state it in writing. Otherwise they will be required to return it, trackable, at their expense.

Which could be a bit hard as perfume is classed as a dangerous good. I assume you have a dangerous goods licence, otherwise you wouldn't have posted it in the first place.