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"why would people be closing short positions if the rumor was that the firm was going belly up? "
"Those with short positions want to see a share price fall and fall and fall. The more it falls the more they stand to make."

You might want to close before it goes belly up so you get your money now and not 18 months or years later. if you already have 90% of the maximum gain realisable now, it's not worth tying it up waiting for the last 10%.