Open Series 7 sponsorship window then leaving firm

Hello everyone hope you are all doing well.

I have to leave my current firm for a short time for some personal reasons and will be taking about 2 to 3 months off to recover.

My firm has offered to open up my sponsorship window to take my series 7 but I will have to leave before I actually sit down to take the test. I have already passed my 6 and 63

My question is can I still take the test in a month if I am not with the firm that opened my window/sponsorship?

If I can take the test, do I just present my passing test to my new firm when I come back to the workforce? Thank you everyone in advance.