Asymptotic conjecture for subword complexity sequences over a k-letter alphabet

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

$$a_k(n) \sim \log_2(k) \times 2^{n/2}.$$

The conjecture is motivated by numerical data and generalizes the binary asymptotic prediction to arbitrary alphabet size; its resolution is not given in the source.