Finite-length augmentation of minimal Dyck rationals to represent real numbers

Let $\mathcal{D}_{r_{\text{min}}}$ be the language of minimal Dyck-word representations, and consider augmenting its words with additional notation or productions. Finite-length real-representation conjecture. It is possible to augment $\mathcal{D}_{r_{\text{min}}}$ so that words of finite length represent the set of real numbers. The conjecture proposes extending the existing finite-word representation of $\mathbb{Q}$ to $\mathbb{R}$, possibly by a vinculoid or markup mechanism, but does not specify the form of the augmentation.