The Texan conjecture on dimensions of bounded-type continued fraction sets

For each non-empty finite set $\mathcal{I}\subset\mathbb{N}$, let

$$E_\mathcal{I}=\{x\in(0,1): a_n(x)\in\mathcal{I}\quad\forall n\in\mathbb{N}\},$$

where $a_n(x)$ are the continued-fraction digits of the irrational number $x$. Texan conjecture. The set

$$\{\dim_\mathrm{H} E_\mathcal{I}:\mathcal{I}\subset\mathbb{N}\;\text{finite}\}$$

is dense in $[0,1]$. This conjecture concerns the possible Hausdorff dimensions of bounded-type continued fraction sets and asserts that finite digit restrictions realize dimensions arbitrarily close to every value in the unit interval.