The density limit conjecture for the binary Ulam set

Let $\mathcal{W}_{2,n}$ denote the set of binary words of length $n$, and let $V_n=\mathcal{U}(\{0,1\})\cap\mathcal{W}_{2,n}$. The density of the binary Ulam set is measured by $|V_n|/|\mathcal{W}_{2,n}|=|V_n|/2^n$. Density limit conjecture. There \exists some $0<r<1$ such that

$$\lim_{n\to\infty}\frac{|V_n|}{|\mathcal{W}_{2,n}|}=\lim_{n\to\infty}\frac{|V_n|}{2^n}=r.$$

The claim formalizes the observed convergence of the computed densities of binary words in the Ulam set; the source gives no proof or resolution.