This is just a weaker version of the twin prime conjecture. You’re asking for primes that differ by 2^k instead of 2. Last I heard the best result was from Polymath 8a.
Big side note, but what you just exhibited is one of my favorite things about this community: admitting being wrong.
It wasn’t really until the workplace where I encountered level-headed people that would just shrug their shoulders and say “You are right. I was wrong” and then move on.
To take it a little further, what I was even more specifically interested in was a sequence of primes following the principle of a gray code [1] where each subsequent prime would differ from the previous prime by precisely one bit.
So basically, what is the longest sequence of primes you could find following that pattern?
http://michaelnielsen.org/polymath1/index.php?title=Bounded_...
Edit: Actually I was wrong, differing by a binary digit is much stronger than a difference of 2^k. But that’s a starting point.