No, Pearson won't ask for a proof of your eligibility. From what I know, they are the ones who would contact CTS if you are eligible. Hence, after Pearson confirms to CTS that you are eligible, then they will send you your ATT.
Here's how it works..
--you have 3 years (from the date CTS receives your application) to complete your IL Licensure. which means, within 3 years, you have to take and pass your NCLEX.
--The eligibility letter, CTS will send you that letter if you completed all their requirements (CES, Toefl--if needed, Fingerprint card etc) and found eligible.
The letter has has no "expiration" (compared to let's say California, their eligibility letter is valid for a year) BUT you have to remember that your whole application for IL is valid only for 3 years.
---If you are now eligible, you can now register to pearson vue. Then, they will send you your ATT. your ATT is valid only for 90 days..
My advice for you, don't register yet for pearson vue if you are not yet ready to take the NCLEX 90 days from now so they won't send you your ATT. Do you get me? but of course, don't take too long as well since as I've said the whole application process (taking and Passing the test) lasts for 3 years only.