Another variant would be a Box where you have to open all Locks at once, or where you cannot pull out a key as long as the Lock is open!
Then you would only 3 Locks, all of them can be opened with the same type of key:
The Vizir has 2 Standard Keys
The Slaves each have one 1 Standard Key
If three Keys come together you open the box!
Another way, which is also kind of trickey, but won't break any of the rules:
The Box has 3 Locks: 1,2,3, The locks are attached in an even triangle, right next to each other and the locks are designed in a way, you can only remove the key when the lock is closed. As long as it is open the lock is stuck.
Each Key has a big nice cylindrical grip. So Big that you cannot put in two keys at once, if they have the same length, as their grips will hinder each other. So you can only insert 3 keys, if all 3 keys have different lengths: You insert the shortest first, then the second and the longest last.
Each slave now gets 3 Keys (one for each lock) but each slave will get a set of 3 keys which are all of the same length (Slave 1 has 3 keys of length 1, Slave 2 three keys of length 2, ...) So although each slave has a key for all 3 locks, they can only use one of them at once.
When 3 Slaves come together, they have 3 different lengths among them, so they can open the Box. The Vizir has 2 Keys: 1 and 2 with Length 5 and 6, so they can combine with a Key-3 form any Slave!