Thomas-Kilmann model is widely misunderstood.
First, it’s a behavioral model, not a guide to conflict resolution. Then, it has nothing to do with win-win, lose-lose etc.
The only possible positive solution for both parties of the conflict lies in collaborating area. When both input high levels of assertiveness and cooperativeness. “Growing the pie”.
Competing provokes deeper conflict. My way or the highway approach, when submission by force is in no way a solution at all. Playing the muscles is a “lose-lose” too.
And finally, compromising, the cornerstone of modern European ideology is nothing but “lose-lose”.
An interesting book on this subject was written by the FBI hostage negotiator Chris Voss, “Never Split the Difference”. I recommend.