Can the characters wager in with collateral items more than once?

Each of the characters have a signature item that they may 'buy-in' with at the start of a tournament. Knocking them out will earn you that item as a playable/wearable item in Team Fortress 2.

Is it possible to have these encounters again, where the characters wager in with collateral? If so, do you receive the item again should you beat them?

I've been over the Unlockables and the Team Fortress 2 Items pages on the Poker Night Wiki but this isn't explicitly spelled out either way.

Solution 1:

After you beat them, they will not wager in again with that item. If you lose, they will wager in with it again, and usually soon.

But if you already have the item, it won't come up again. I can confirm this personally, can't find a link for it though, but I have many hours put into that game after getting all the items, and I've never seen them appear again.