The first signed path conjecture for finite binary trees

Let $D$ and $R$ be finite, binary trees with the same number of leaves. A sign assignment assigns a sign to the relevant vertices of a tree, and a word of rotation symbols valid for $D$ is a sequence of signed rotations that can be applied successively to $D$. Write $Dw$ for the tree obtained after applying the word $w$ to $D$.

First signed path conjecture. For every pair of finite, binary trees $(D,R)$ with the same number of leaves, there is a sign assignment of $D$ and a word $w$ of rotation symbols valid for $D$ such that

$$Dw=R.$$

This conjecture asserts that every pair of finite binary trees with the same number of leaves can be connected by a valid signed rotation path. The surrounding results establish one direction relating valid signed paths to color vectors, while this converse is presented as a conjecture and is later restated in the paper.