The duplicate-leaf parse-word conjecture for binary trees

Let $T_1'$ and $T_2'$ be $(n-1)$-leaf binary trees. Let $1 \leq i \leq n-1$, and let $T_1$ and $T_2$ be the $n$-leaf trees obtained from $T_1'$ and $T_2'$ by duplicating leaf $i$. A parse word is a word that parses both trees according to the paper's labeling rule. Duplicate-leaf parse-word conjecture. There exists a parse word $w = w_1 \cdots w_n$ of $T_1$ and $T_2$ such that $w_i = w_{i+1}$. This would reduce the search for parse words by showing that duplicated leaves can receive the same letter, but the supplied text gives no resolution.