Follow up or wait it out?

I came across a lateral analyst position for a mid-tier BB on their career portal and given I don't have any alumni at this place I cold emailed an HR person and a FO VP, both of whom responded to me. Had a call with the HR person that went okay, and had a quick drink with the VP that also went okay. The VP works in the division that is recruiting (although he says he is not directly involved in it) and offered to follow up with the HR person I spoke to. I applied online after speaking to them, emailed them both thanking them for their time, and haven't heard back in roughly a week. My application status online is also "under review".

The HR person said there wasn't an actual schedule for first/second rounds so my impression is they aren't in a rush to hire anyone. At this point what is the best thing to do? Should I ping them again after 1-2 weeks? Should I just wait and hope they get back to me at some point? Should I start reaching out to more people? I'm open to more aggressive tactics but want to be smart about it.