Huang's conjecture on local obstructions in Zaremba theory

Let $A\subset\mathbb N$ be a finite alphabet and let $X_A$ be its limit set. For each $m\in\mathbb N$, define reduction modulo $m$ by

$$\pi_m\colon\mathbb Z_+\to\mathbb Z_m=\{0,1,2,\ldots,m-1\},\qquad \pi_m(q)=q\pmod m.$$

The alphabet $A$ has no local obstructions if, for every $m\geq1$, the set of denominators of reduced rationals represented by finite continued fractions with digits in $A$ has image $\mathbb Z_m$ under $\pi_m$. Huang's conjecture. If

$$\dim_H(X_A)>\frac56,$$

then $A$ has no local obstructions. This predicts that sufficiently large Hausdorff dimension rules out every congruence obstruction to the Zaremba-type denominator problem; the supplied text does not state a resolution.