Context-free grammar approximation conjecture for irrational numbers

Let $x$ be an irrational real number. Let $G_x$ be a context-free grammar that generates an infinite sequence of Dyck rationals $(s_i)_{i\in\mathbb{N}_{+}}$, and let $\upalpha_{\mathbb{Q}_{r}}$ denote the interpretation map from Dyck-rational words to rational numbers. Irrational-approximation grammar conjecture. For every irrational number $x$, there exists such a context-free grammar $G_x$ for which

$$\upalpha_{\mathbb{Q}_{r}}(s_i)\longrightarrow x\quad\text{as }i\longrightarrow\infty.$$

This is an alternative to representing irrational numbers exactly by finite words: the grammar would generate successive rational approximations following a pattern that converges to the chosen irrational number. The source gives no resolution of the conjecture.