The second signed path conjecture for pairs of binary trees

Let $D$ and $R$ be finite, binary trees with the same number of leaves. An edge path in the associahedron from $D$ to $R$ is sign consistent when its sign structure is balanced; equivalently, its rotations admit a compatible sign assignment throughout the path.

Second signed path conjecture. For every pair $(D,R)$ of finite, binary trees with the same number of leaves, there is a sign consistent path from $D$ to $R$.

The source states that this is equivalent to the first signed path conjecture via the sign-consistency theorem. The supplied text gives no resolution, so the conjecture remains open.