Non-empty core conjecture for correlated gift exchange coalitions

Let $P$ be the set of players and let $\mathcal{K}$ be a coalition structure, with core stability meaning that no subset $S \subseteq P$ can deviate to form a new coalition and make all members of $S$ strictly better off. Assume standard gift exchange rules and correlated valuations. Non-empty core conjecture. The coalition formation game has a non-empty core. In particular, family-sized coalitions $K$ with $|K| \in \{2,3,4\}$ among players with correlated preferences are stable. The claim concerns the stability of coalition formation under correlated preferences; the supplied text does not state whether it has been proved or identify conditions beyond standard gift exchange rules.