Apparently he needs to hold those options until 06/21.
No. No one ever has to hold options till expiry. Most don't. The longer you hold options, the more you lose to time decay. And if he does decide to exercise, he'd have to come up with $240 million, which would mean selling a lot of his stock, with predictable results.
Gill does have a problem: he has so many calls--120,000 contracts--that if he dumps too many, he'll drive the price WAY down, Yesterday, the last trade in those calls was priced at $9.75. On Thursday, they'd closed at $27.02. Reportedly, Gill bought his at about $5, so his position has already taken quite a hit.
People on the GME board are saying they think he "knows" what Cohen is planning, and so he's hanging on. I very much doubt that, because I don't think Cohen wants to go to prison on insider trading charges.