The double-zero parse-word conjecture for path trees

Let $n \geq 4$. A path tree is the path-tree object used in the paper, and a parse word is a word that parses a binary tree. Double-zero parse-word conjecture. Every pair of $n$-leaf path trees parses a word of the form $u00v$ for some possibly empty words $u$ and $v$. The statement has been explicitly verified for $n \leq 12$, while the source notes that it does not hold for general binary trees; no general proof is supplied.