Keeping the Conversational Score : Constraints for an Optimal Contextualist Answer?

Conversational contextualism states that the truth-conditions expressed by knowledge-attributing sentences vary relative to the context of utterance. This context is determined partly by different standards the person involved must meet in order to make the sentence true. I am concerned with the question of how these standards can be raised or lowered, and especially what happens to the standards and the conversational score when parties in a discussion push the conversational scores in different directions. None of the available options for an answer seems satisfying. I argue that this results from a misunderstanding of the characteristics of the situation at hand.