What is stopping Melvin Capital from lying about closing their position in order to deflate the stock price so they can actually close out their short position when they are due this coming Friday?

Yeah that is basically it. If you are safe why report on it? You don't gain anything. If you are in danger then a little lie doesn't hurt because even a tiny chance of success could mean billions of dollars for you.

Couldn't they close their position and still have the short interest go up because of other investors opening new positions?

If these institutions are going on record as closing their positions, when they actually havent, aren't they committing some sort of fraud?

If you follow this in more detail you will see they have already misled the public on their positions several times. But when the choice is between your fund’s bankruptcy and an SEC fine, people do crazy things.

Also the borrow fee for shorts is insane now, like 23%.

Annualised? If so, that's peanuts.