Yes. Your task runs on Tycoon's side, not inside your browser. Closing the tab, putting the laptop to sleep, switching to your phone, or signing out does not stop work that is already underway.
Do I need to keep the app open for it to make progress?
No. Nothing about the task depends on a live window. An open page only refreshes what you see; it does not drive the work.
What happens to everything I miss while I'm away?
Nothing is lost. Progress, evidence, and the final result are written to the task itself, so the task thread is a complete record when you come back. Open it from Tasks and read on from where you left off.
Can I start something else while it runs?
Yes. Long work does not lock the product. You can chat with Tycoon Agent, delegate other outcomes, and review earlier tasks while it continues.
So what does stop a task?
Not a closed window. A task stops when you cancel it, when the wallet runs empty — that one does interrupt work already under way, until you top up — or when the platform hits a failure it cannot get past on its own. See How to cancel a task and What happens when the wallet runs out.
For what each state means while work runs, see Task states and what each means. For how you find out it finished, see How will I know when a long task is done?.