Exponential-growth conjecture for $k$-mer minimizer counts

Let $\Sigma$ be an alphabet, let $m\geq 1$, and let $w\in\Sigma^m$ be an $m$-mer. Write $\pi_k(w)$ for the number of objects counted by the paper's minimizer-count function, and let $\max(\Sigma)$ denote the maximum symbol of the alphabet. Exponential-growth conjecture. If $w$ does not start with $\max(\Sigma)$, then

$$\pi_k(w)\sim\alpha(w)\cdot|\Sigma|^{\beta(w)k}$$

as $k\to\infty$, for some constants $\alpha(w),\beta(w)>0$. This conjecture formalizes the observed exponential growth of $\pi_k(w)$ for examples whose first symbol is not maximal; the supplied context gives numerical regression evidence but no proof or resolution.