Conjectured recurrence for differences of subword complexity sequence counts

Let $a_k(n)$ denote the number of distinct subword complexity sequences of length $n \ge 1$ over a $k$-letter alphabet. Recurrence conjecture. There exists a function $f(k)$ such that, whenever $n \le f(k)$,

$$a_{k+2}(n)-a_{k+1}(n) = a_{k+1}(n-1)-a_k(n-1).$$

This conjecture arises from numerical data and proposes a recurrence for successive differences in the counts of subword complexity sequences; the source does not establish it or indicate a resolution.