I thought that at first as well. Then I read the notes which made me reframe it as ‘odds your digit sequence won’t include a six ever’ and note that checking up to 2^50000 has only two candidates with the first 15 digits even, and I came down on ‘shrinking so quickly it’s super unlikely’. No proof here due to HNs comment limits of course..
I wonder if we can get a sense of how fast it would grow if we hypothesize it is an infinite sequence.
And if it is a finite sequence, one could define f(p, n) as the sequence of successive exponents of 2 such that the ratio of even digits over its total number of digits is greater than p. This could be an interesting way of describing a set of fast growing functions from exponential growth (p=0) to arbitrarily fast growth as p grows closer to 1 (or P where P is the smallest number such that f(P, n) is a finite sequence).