The problem I'm seeing is with Case 2 of Lemma 4.5 in the latest version of the ArXiv preprint:
Unallocated contested piece a has proper trims on it (recall that we are ignoring part that is left of the rightmost non-winner’s trim mark). Since the only proper trim on a piece can be by some agent in W, this means that a admits a proper trim by an agent j ∈ W. But this is a contradiction because j finds a strictly more preferred than his current allocation.
It isn't obvious to me why this is true. As I mentioned in a blog post of mine on this topic:
This last sentence is based on the following observation: any trims by some agent j ∈ W must be such that the right hand side is equal in value to j’s most-valued unallocated piece. However, during the next iteration of the while loop from the Subcore protocol, it is possible that all of the contested pieces that j values as much as any of the unallocated pieces from the previous round have been allocated to other agents in (now expanded) W. Once this happens it is possible for agent j to place trim to the right of the agent who this case 2 from the lemma incorrectly is claiming must continue having the right-most trim.
Can someone explain what either I or the authors are missing here. I want to understand this.