L3 to L4 time at G

For a very strong L3 hire and 5.5 YOE, how much time will it take to get a promotion? Is 6 months target realistic? PS: Downlevelled from L4 to L3. Desperate to become L4. I am thinking of talking to a manager before I join G and tell him my expectations, and ask him to assign me responsibilities of L4 from the beginning itself. Thanks.