Unilateral termination without grand-state absorption

Prove Theorem (arrival under the preference condition) under unilateral termination without assuming that every grand state is absorbing, thereby removing the residual off-path configuration identified in Remark on unilateral termination.

Background

The paper proves arrival under a static payoff-separation condition when every grand state is absorbing. Under unilateral termination, however, only grand states that are actually entered are shown to be absorbing. The remaining possibility is that an unentered grand state has off-path continuation values low enough that termination from that state makes the merge-all move dominated at its parent, sustaining a non-grand recurrent class.

Resolving this problem would establish whether the preference condition alone guarantees eventual formation of the grand coalition under unilateral withdrawal, without the additional absorption hypothesis required by the current theorem.