Hensley's dimension-denominator conjecture for continued-fraction alphabets

Let $\mathcal A\subset\mathbb N$ be a finite alphabet. Let $\mathcal C_{\mathcal A}$ be the continued-fraction Cantor set

$$\mathcal C_{\mathcal A}=\{[a_1,a_2,\dots]:a_j\in\mathcal A\text{ for all }j\ge1\},$$

and let $\mathcal D_{\mathcal A}$ be the set of denominators of the partial convergents to $\mathcal C_{\mathcal A}$. Denote the Hausdorff dimension of $\mathcal C_{\mathcal A}$ by $\delta_{\mathcal A}$.

Hensley's conjecture. The denominators contain every sufficiently large natural number if and only if the Hausdorff dimension exceeds one half:

$$\mathcal D_{\mathcal A}\supset\mathbb N_{\gg1}\qquad\Longleftrightarrow\qquad\delta_{\mathcal A}>1/2.$$

The source later states that this claim is false: the alphabet $\{2,4,6,8,10\}$ has dimension $0.517\dots>1/2$ but has congruence obstructions. Thus the conjecture is refuted.