Hensley's dimension criterion for Zaremba's conjecture

For an integer $A\geq 2$, let

$$E_A=\{x=[0;a_1,a_2,\ldots]\in[0,1]:a_j\leq A\},$$

and let $\delta_A$ be the Hausdorff dimension of $E_A$. Define

$$D_A=\left\{N\in\mathbb{N}:\frac{a}{N}\in E_A\cap\mathbb{Q}\text{ for some }a\in(\mathbb{Z}/N\mathbb{Z})^\times\right\}.$$

Hensley's conjecture. One has $\delta_A>1/2$ if and only if $D_A\supset\mathbb{N}_{\gg 1}$, meaning that Zaremba's conjecture holds for all sufficiently large $N$.

This proposes a precise link between the Hausdorff dimension of the bounded-digit continued-fraction set and the eventual validity of Zaremba's conjecture. The paper describes it as a claim of Hensley; its resolution status is not specified in the supplied text.