Dinei Mamonus Shaila on Shared Lottery Winnings

This is a strictly halachic shaila (not a secular law question.) If a group of colleagues jointly purchase a set of lottery tickets, with one person being in charge of buying the set of tickets for the group. The “gabbai”, in addition to purchasing the (say) 100 tickets for the group, purchased a few extra tickets for himself from his own money. (This is provable by the fact he is holding more tickets than the group purchased.) Unfortunately, prior to the drawing the group never asked or received a list of which ticket numbers were jointly owned by the group.

If one of the tickets wins the jackpot, and the gabbai claims he kept a seperate cheshbon of which were the groups tickets and which were his, and the jackpot was won on one of his privately purchased tickets… Who had the burden of proof? Does the group have to prove that the winning ticket was theirs — a task that is probably impossible for them since only the gabbai kept the cheshbon? Since, after all, the gabbai is the one holding the winning ticket and the group is demanding it, therefore they have the halachic burden of proof?