Vanishing leading asymptotic coefficient for fixed-occurrence word enumeration

Let $k,r\in\mathbb N$, let $v\in[k]^*$ be a pattern using $d>1$ distinct letters, and let $f_r^v(k,n)$ count words in $[k]^n$ containing $v$ exactly $r$ times. Let $M_r$ be the maximal number of states with $d-1$ loops in a simple path in the automaton $Au(v,r,k)$. For constants $K_r$ determined by

$$\lim_{n\to\infty}\frac{f_r^v(k,n)}{n^{M_r}(d-1)^n}=K_r,$$

vanishing-coefficient conjecture. There exist $k,r\in\mathbb N$ and a pattern $v\in[k]^*$ such that $d>1$ and $K_r=0$. In that case, there exist $L_r\in\mathbb N$, $L_r<M_r$, and $\widetilde K_r\in(0,\infty)$ such that

$$\lim_{n\to\infty}\frac{f_r^v(k,n)}{n^{L_r}(d-1)^n}=\widetilde K_r.$$

The authors report that they observed $K_r>0$ in many examples, so the conjecture predicts an exceptional pattern and a lower polynomial exponent in its asymptotics; no resolution is supplied.