While I agree with @DavidEspina (as usual), it might be useful to explore alternatives.
You hint that the underlying cause is a conflict of personalities/egos ("showing weakness") I abstain on the resolution of these conflicts, except to say that they're going to be messy and unpleasant.
But another possible source of conflict might be an unvoiced, underlying disagreement. My first step would be to listen to both of the experts, but I would be tempted to ask them both to drop the question on the table, and instead to ask:
If a third party were to suggest another approach - one that is not related to either of your approaches - how should we measure that approach? How would we judge if it was successful? How would we estimate the cost & benefits?
There is a chance that the disagreement is not the one over which they are arguing, but a more fundamental disagreement about what "successful" means, or about which feature to optimize.