Infinitely many lengths with exactly $2^e$ very odd sequences

For each natural number $n$, let $S(n)$ denote the number of very odd binary sequences of length $n$. For an integer $e\geq 1$, consider the level set of lengths satisfying $S(n)=2^e$. The infinitude conjecture. For every $e\geq 1$, there are infinitely many integers $n$ such that

$$S(n)=2^e.$$

This is proposed as a strengthening of the known fact that each such level set is nonempty. The paper explains that the claim would follow from the existence of infinitely many suitable non-Wieferich primes, a statement related to Artin's primitive root conjecture and currently unproved.