I think ANT is confusing shorts with naked shorts. What he is saying sounds correct for normal shorts. A naked short is when a MM sells stock that they don't have yet. It is kind of like I'll sell you a hamburger today that I will have tomorrow, even though I don't have it today.
When a naked short position needs to cover due to new cusip for example, the MM must find shares to cover the sale of shares that they made when they didn't actually have shares. No more "I'll charge you today for a hamburger tomorrow". They will need to buy a hamburger from someone else if they don't have one in hand to cover. Because of this, there can be a lack of shares so there can be a scramble to find shares. Since the MM's have to eventually cover, they will be forced to pay whatever the going rate is to cover.