Is it necessary to remove all 4 wisdom teeth?

If you don't get the teeth removed you run a real risk of a relapse in your orthodontics. Instead of being ready for a retainer you would have to redo much of the tooth repositioning again. Also in your picture, note the decay and erosion on the tooth in front of the partially impacted wisdom tooth. I'm sure you don't want that. So it sounds like having had your 4 bicuspids removed ,you still don't have enough room for your wisdom teeth to come i. You must have small jaws. Follow the advice of professionals and don't listen to people who do not understand dentistry. I wish you good luck.