What are some ideas concerning the prompt, "Staying true to our core values is the best way to deal with conflict"?
Rather than core values being the moderator of conflict and conflict resolution, it may be more accurate to express the concept differently. Values may or not be laudable. A mercenary person values greed. Greed is not a very useful value in conflict as greed may provoke conflict! Maybe it is more useful to think in terms of something that supersedes individual values. Maybe it is useful to think in terms of traditionally socially honored values of respect and civility. In this case you might say: Staying true to society's core values of civility and respect is the best way to deal with conflict.
This is only true, however, if our core values are good ones for the situation at hand. If our core values are bad, then staying true to them is counterproductive. To look at one of the examples above, many of the people of Maycomb stayed true to their core values, which were that black people were inferior and should be kept down. This was not (at least not in our minds) the right way to deal with conflict.
We can see examples of this in the real world as well. To take a really extreme example, Hitler's core values were racism and a desire to dominate. He stayed extremely true to those goals and we all know the result.
So, simply sticking with core values is not always a good thing if those core values are wrong.
Since you posted this in the "Literature" section, I will answer your question using examples from literary texts.
First, one could look at the character of Macbeth. In the beginning of the play, Macbeth was renowned for his bravery and loyalty. Through his actions, the murder of Duncan, one can see that he forces conflict upon himself (his actions result in his death). If Macbeth would have stayed true to the man he was prior to hearing the witches' prophecy, Macbeth would have gained the crown through noble acts and forgone the conflict.
Another example of maintaining core values can be seen in To Kill a Mockingbird. The main character Atticus Finch must defend a black man accused of rape. Given that the crime happened in a largely white community, Tom Robinson is basically found guilty before the trial has even began based upon prejudice. It is through Atticus' honesty to his core values that he is able to maintain the influence he has over his children, Scout and Jem.