No! At the start of a new streak, you will be able to reroll through every item in the weekly prize pool. Any item you skip while rerolling will be removed from the list and will not appear again until you have cycled through the entire prize pool, at which point you will be able to cycle through all the prizes again. That should make it easier to find the item you want!
Please note! If you do not complete your quests for a day after the prize pool has been updated, then your prize will change to a prize from the new pool and your re-roll cost will reset. You will be unable to reroll items from the previous pool once the new one is live.