Path conjecture for preorder access in binary search trees

Let ${\mathcal{A}}$ be an online binary search tree algorithm. Starting with any initial tree $T$ with $n$ elements, let $X$ be the preorder sequence of a binary search tree $T'$ that is a path. Path conjecture. The cost of accessing $X$ is $O(n)$. The source states that this conjecture is implied by the deque, traversal, and split conjectures, making it an easier common consequence; its status is not resolved in the supplied source.