I think there's confusion between the idea of a "tab" in Teams versus a "tab" in the browser. In the browser, you can have a tab open, but be active on another tab. In that case, a websocket could be running, updating the content so that when a user goes back to the tab, it's already updated.
In contrast, in Teams, when a user views a tab, it's effectively loaded into an iframe. When the user browses away, the iframe is destroyed, and when they come back, the iframe it recreated, and the page -completely reloaded- again. As a result, there is no need, and it is not possible to "push" data to the tab when it's not in focus. However, when it is in focus/being used by the user, you can certainly keep it up to date using, for example, signalr.
