Positive-density conjecture for words avoiding forbidden signatures

Let $\pmb{\omega}=(\omega_0,\omega_1,\ldots,\omega_k)\notin\mathcal{F}$ be a finite-length word that contains no $\pmb{\tau}\in\mathcal{F}$ as a subsequence of consecutive symbols. Positive-density conjecture. The limit

$$\lim_{n\to\infty}\frac{\left|\left\{M\leq n:\;\|M-j\|_\infty=\omega_j\text{ for }j=0,1,\ldots,k\right\}\right|}{n}$$

exists and is strictly positive. This is the formal version of the paper’s stronger density claim for admissible words. Its status is not resolved in the supplied material.