Context-free grammar for minimal recursive prime factorizations

Let $\mathcal{D}_{r_{\text{min}}}$ denote the language of minimal Dyck-word spellings for the recursive prime factorizations of natural numbers. A context-free grammar has nonterminal symbols $N$ and $S$ and generates words over the parenthesis alphabet, with productions

$$N\rightarrow \varepsilon,\qquad N\rightarrow,\qquad N\rightarrow S,$$ $$S\rightarrow SS,\qquad S\rightarrow S,\qquad S\rightarrow (S),\qquad S\rightarrow ().$$

Minimal-spelling grammar conjecture. The language underlying $\text{RPF}_{\mathbb{N}_{r_{\text{min}}}}$ is generated by this context-free grammar. The grammar is ambiguous, with two shift-reduce conflicts in an LALR parser; resolving both conflicts in favor of shifting was used to verify the minimal spellings of the first thousand natural numbers. The parser evidence does not establish correctness for all natural numbers.