The path-tree parse-word and level-restriction conjecture

Let $n \geq 4$, and let $T_1$ and $T_2$ be $n$-leaf path trees such that leaf $1$ is on level $1$ in $T_1$ and leaf $n$ is on level $1$ in $T_2$. A pair is mutually crooked if it cannot be obtained by duplicating a leaf in a pair of $(n-1)$-leaf trees, and weakly mutually crooked if it cannot be obtained by triplicating a leaf in a pair of $(n-2)$-leaf trees. A parse word is a word that parses both trees. Path-tree parse-word and level-restriction conjecture. The following assertions hold: (i) if $T_1$ and $T_2$ have no parse word of the form $00v$ or $v00$, then they have a unique parse word up to permutation of the alphabet; (ii) if they have no parse word of the form $00v$ and are mutually crooked, then they have a parse word of the form $01v00$; (iii) if they have no parse word of the form $00v$, then the only possibilities for the pair of levels of leaves $2$ in $T_1$ and $n-1$ in $T_2$ are $(2,3)$ and $(k,2)$ for some $k \geq 2$; if they are weakly mutually crooked, one has $2 \leq k \leq 4$, and if they are mutually crooked, one has $2 \leq k \leq 3$. These claims are proposed as tools toward proving that every pair of binary trees has a parse word; the supplied text gives no resolution.