Now I'm curious about that last result discussed, for bases less than 10 - "proving that there are infinitely many primes that don’t have any 7s (or any other d

Now I'm curious about that last result discussed, for bases less than 10 - "proving that there are infinitely many primes that don’t have any 7s (or any other digit you might choose)". It sounds like Maynard decided to stop at proving the result for all bases 10 and above, but could it be true for the smaller bases too? In the case of binary numbers, proving that there are infinitely many prime binary numbers without a 0 is the same as proving that there are infinitely many Mersenne primes, so the (non-trivial) binary case seems to still be a difficult open problem. It would be fascinating if there's a cutoff in base expansions where this stops being true.