This is intentionally reported as bug and not as a feature request.

Use case: Test Engineer A and Test Engineer B have opened a Test Run. Test Engineer A assigns test cases to himself, and Test Engineer B wants to assign all the unassigned test cases to himself. Test Engineer B doesn't have an idea that Test Engineer A also wants to assign something and in fact overrides all the changes. In order to not have this race condition, each Test Engineer should refresh the page manually to see the most recent status

Idea: Introduce event-like logic on Test Case Status and Assignee Update. When something is updated, asynchronously update this object on the page for other users who stay on the same page (in Test Run and in Execution Wizard)