Bourgain–Kontorovich's revised Hensley conjecture with admissibility

Let $\mathcal A$ be a finite alphabet, let $\mathfrak D_{\mathcal A}$ be its denominator set, let $\mathfrak U_{\mathcal A}$ be the set of admissible integers, and let $\delta_{\mathcal A}$ be the associated Hausdorff dimension. Bourgain–Kontorovich's revised Hensley conjecture. If $\delta_{\mathcal A}>\frac12$, then $\mathfrak D_{\mathcal A}$ contains every sufficiently large admissible integer. The source presents this as the appropriate revision of Hensley's original claim in the presence of congruence obstructions; it is not resolved in the supplied text.