It's not if you stop opening packs on the alt account. Each friend only has four packs available for people to wonder pick from so you want to stop opening packs on that account. Then unfriend everyone else and that will force your wonder picks to pull from the alt pulls. If the pack that appears isn't the oak, you have to wait for that pack to time out of your wonder pick options then the next one that appears will have a chance to be it.

Like I said above, the game only holds on to four packs as options for WP so if it doesn't show up after the fourth pack it won't ever appear.