Some sets don’t restrict to a class. And the quest doesn’t require to use any class-specific skill. So, it is not a bug. In fact, it was known many years ago.
In the Years of Wars guide of Icy-Veins (in 2017), it said:
Since you need to grind out two full other sets from another class, we highly recommend that you team up with players of another class; early in a season upgrades come frequently, and you can easily finish this conquest with subpar duplicate pieces your friend throws away.
this suggestion implies the conquest can be done with one class.