Linear growth and repeated values of the automaton exponent $M_r$

Fix $n,k\in\mathbb N$ and a pattern $v\in[k]^*$ using $d>1$ distinct letters. Let $M_r$ be the maximal number of states with $d-1$ loops in a simple path in the automaton $Au(v,r,k)$. Conjecture on $M_r$. (a) The limit

$$\lim_{r\to\infty}\frac{M_r}{r}$$

exists and belongs to $(0,\infty)$. (b) There exist $k\in\mathbb N$, a pattern $v\in[k]^*$ with $d>1$, and an increasing sequence of integers $(r_n)_{n\in\mathbb N}$ such that $M_{r_n}=M_{r_n-1}$. The preceding asymptotic theorem shows that $M_r$ is non-decreasing in $r$; the conjecture predicts both a positive linear asymptotic rate and infinitely many possible repetitions, but the source provides no resolution.