Existence conjecture for growth rates of real-parameter restricted words

For each positive real number $r$, let $\mathcal{W}_{r,n}$ be the set of restricted binary words of length $n$ with parameter $r$. Existence conjecture. For given $r \in \mathbb{R}^+$, the limit

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

exists. For positive rational parameters, the corresponding limit is obtained from the rational generating function; the conjecture extends the question to all positive real parameters and is unresolved in the supplied source.