Enayati and Green's asymptotic conjecture for binary subword complexity sequences

Let $a_k(n)$ denote the number of distinct subword complexity sequences of length $n \ge 1$ over a $k$-letter alphabet. Enayati and Green's conjecture.

$$a_2(n) \sim 2^{n/2}.$$

This conjecture predicts the asymptotic growth of the number of binary subword complexity sequences; the source provides numerical data but no resolution.