The proof is nearly derisively brief: the player simply needs to reduce their declared utility on every other admissible pure outcome, until these are no longer admissible. Then O would be the only outcome available to the bargaining solution.

This is a most excellent proof (it makes everything obvious) and I am convinced by this that agents should not insist that their outcomes above all lie on a Pareto frontier, because some of the points on a Pareto frontier represent unfair splits of gains from trade a la Ultimatum. An auction system between multiple agents might resolve this, but meanwhile one must be willing to refuse certain proposed 'Pareto improvements' in order to possess any negotiating power about which Pareto improvement to make, and it may also be that by the nature of utility functions it is not possible to distinguish aliens that have cleverly rejiggered their utility functions from aliens which were born with them.