Bermond's lobster conjecture

Let $T$ be a tree, and let $P$ be a longest path in $T$. A tree is 2-distant, or a lobster, if every vertex of $T$ has distance at most $2$ from $P$. A tree is graceful if its vertices can be labeled uniquely from $0$ to $n-1$ so that its edge weights are the set $\{1,2,\dots,n-1\}$. Bermond's lobster conjecture. Every lobster is graceful. The conjecture is a strengthening of the known graceful-tree results for paths and caterpillars. The paper proves the claim for lobsters with a matching covering all but one vertex, but the conjecture for arbitrary lobsters remains open.