What is the nub of the dispute here? Is it the applicable law, or is it different views of the facts (or both)?

Even where you have explained that you don’t want to raise neutrality concerns of opposing counsel by learning case details in the initial call, it is helpful to know whether the dispute is a battle over facts, or law, or both.  Or something entirely different.

If you and opposing counsel were negotiating, without any involvement by the clients, do you think you could settle it, without need for mediation?

It’s so much more elegant than just asking if there is a client problem, and may prompt a richer response. This question asks the attorney to make an educated guess about how much of opposing counsel posture is real as well as his or her own position – and how much is negotiation puffery or clients’ demands.  Particularly where the lawyers have worked through significant discovery or motions on the case, they will have an intuitive sense of their counterpart’s approach, whether he or she acknowledged a weakness uncovered in depositions, and the nature of his or her interaction with the client.