Monotonicity and discontinuity conjecture for restricted-word growth rates

Assuming the growth-rate limit exists, define

$$L(r)=\lim_{n\to\infty}\frac{|\mathcal{W}_{r,n+1}|}{|\mathcal{W}_{r,n}|}$$

for $r \in \mathbb{R}^+$. Monotonicity and discontinuity conjecture. The function $r \mapsto L(r)$ is increasing on $[0,+\infty)$ and is discontinuous at every positive rational $r$. This conjecture concerns the dependence of the asymptotic growth rate on the real parameter; the source gives no proof or resolution.