How do you know if they still check your background? If you are relying just on these little statuses on GC Key, then it is a lie. In my GC Key, at the very bottom, it states that my app is approved, I have PPR. However, under eligibility and background it says that they are still checking me (which is impossible at this point). Intake matters, if you are Fall 2020, you should receive PPR soon if your DLI is on the list.
Once again, webform them either a letter of support or enrollment verification if your DLI is approved. If not, then just wait until it is and then remind IRCC about your existence. Send updated LOA only if it changed because for example of an intake change (Fall 2020 to Winter 2022, etc)