Conflict stalemate

Stuck in conflict.

It is inevitable that at times, members get stuck in conflict. They may understand the other members point of view, but still believe that their idea is better. Sometimes, as the leader you need to step in and make the decision. You are now given options and if you have agreed that you are the ultimate decision maker, then you need to decide. It doesn't mean that you are picking favourites (make sure you aren't doing this), but with all of the information at hand, you choose the right route forward. Another member of the team may be given the decision making responsibility, so make sure this is known and understood at the beginning.

And sometimes the issue isn't the issue. 

Sometimes there are other factors that become barriers to resolution. Remember the example of 2 members who had been holding a grudge for 15 years? Those grudges led to one member blocking and fighting for their ideas to "win" and assert power over the other. There are an number of barriers that may exists, your job is to identify them, facilitate discussion and hopefully help resolve the underlying issues. If you don't, your team will stay in dysfunction.

1.  Environment. The discussion is happening in the wrong place, perhaps in an open meeting or office. It needs to be discussed in the right location with the right people. Perhaps the right people aren't in the room, so you need to rearrange to reach resolution. Sometimes the culture, office politics, lack of time or wrong time of day is the issue. So reconvene when everything is right and reach resolution.
2. Relationship. Some people might have unresolved relationship issues that prevent resolution. The conflict is nothing to do with the topic, but something much deeper. It might be a difference in personality style that creates tension, grudges, hierarchy or reputation. A subject matter expert might not feel comfortable challenging the CEO, or someone might be vying for a promotion in another team and doesn't want to rock the boat. They are all putting self-preservation ahead of team results. The leader needs to identify the issue, perhaps talk one to one with each member to mine for truth, then name what is going on and how it is harming the team. Hold the members responsible for their own behaviours and help them move forward by focusing on results.
3. Informational. Perhaps you simply don't have the right information to make a decision. Provided all members are sharing fully and not withholding information, you may need to assign actions to obtain more information, then reconvene. If the information is unobtainable, the decision maker must then decide what level of risk they are willing to take. This is the easiest to resolve.  
4. Individual. Like relationship, this is often harder to solve.  Perhaps one member of the team is a barrier to resolution. They may stop healthy conflict from happening because of their own behaviour, attitudes, beliefs or skills. It maybe IQ, EQ, self esteem, experience or knowledge. They may hold beliefs or values that conflict with the team. They may hold political, social or religious beliefs that are not able to adjust to the teams, and therefore hold the team back. Now, I do believe it is good to have diverse thinkers on the team, but if one members diverse thinking harms the team (homophobia, sexism, xenophobia etc.), you may need to decide who to resolve the individuals issues or needs before moving forward. 

As the leader, your job is not to solve everyone's issues. Your job is to be aware of the potential or existing barriers, name it and then empower the team to either address it or agree to not let it influence the conversation. You are asking members to focus on team results first and holding each member accountable for their behaviours.