Rational generating-function conjecture for maximum independent sets of de Bruijn graphs

For a positive integer $d$ and a fixed diameter $D$, let $B(d,D)$ be the de Bruijn graph and let $a_{d,D}$ denote the number of maximum independent sets of $B(d,D)$. Consider its exponential generating function

$$\sum_{d=1}^{\infty}\frac{a_{d,D}t^d}{d!}.$$

Rational generating-function conjecture. If $D$ is a fixed odd prime number, then this exponential generating function is the ratio of two polynomials, each of degree $D-1$. For $D=3$, the paper proves the analogous assertion with numerator and denominator of degree at most $2$; the conjecture proposes the stated degree bound for every fixed odd prime diameter.