Pinned-spine conjecture for full binary trees

Let $T$ be a full binary tree (FBT) on $n$ vertices. A longest root-to-leaf path is a path $v_0v_1\cdots v_\ell$ with $v_0$ the root. Its alternating extreme spine assignment is

$$s(v_i)= \begin{cases} i/2, & i\text{ even},\\[2pt] n-1-(i-1)/2, & i\text{ odd}. \end{cases}$$

Pinned-spine conjecture. Every FBT $T$ on $n$ vertices admits a graceful labeling $f$ such that, for some longest root-to-leaf path $v_0v_1\cdots v_\ell$, the restriction of $f$ to that path is the alternating extreme spine assignment. This is stronger than ordinary gracefulness because the prescribed spine consumes the largest edge differences. The paper proves the conjecture for comb full binary trees, and reports exhaustive verification for rooted non-isomorphic FBTs through order $23$; the general conjecture remains open.