Did it twice, rankings went up ever so slightly. In fact, visit duration and conversion rates seemed to also increase, but a tiny amount. So small that I'm saying don't do this to get gains, but don't worry about losses from doing it. When we crossed over, the indexing was fast.
Both times, we had empty robots.txt files, no rel= links, and not even a sitemap, so perhaps the slate was clean. Make sure you don't have remnant disallows for https for a bit before you cross over, I think engines could be confused by this (I'm not an SEO).
Disclaimer: I'm a PPC guy, not an SEO guy, I don't use any SEO ranking services, I'm relaying what small glances at data that seemed to be happening, not hard, reliable, torn into stats at all.
Keep an eye on this:
Test mobile speed on an https test page before you go.
If you host videos, the scripts always have tanked us up on protocols, test that too, before.
Knowing you, nothing here I said is of value to you Brett! I do look forward to hearing how it went for you!