The 1089 graph characterization by endpoint digits

Let $N=(a_{n-1}\binom{\!\!\!\!\!\!\!\!\!\!}{\vphantom{1}}\cdots a_0)_g$ be a $(g,k)$-reverse multiple, with endpoint digits $a_0$ and $a_{n-1}$, and let its associated graph be the Young graph. The 1089 graph is the Young graph isomorphic to the $(10,4)$ and $(10,9)$ Young graphs.

1089 graph endpoint conjecture. If there is a $(g,k)$-reverse multiple with $a_0+a_{n-1}=g$, then the Young graph is the 1089 graph. Conversely, if the graph is the 1089 graph, then every reverse multiple satisfies $a_0+a_{n-1}=g$.

The first implication is suggested by the preceding theorem together with the 1089 graph conjecture, while the converse is also expected to follow from that conjecture. The claim is stated separately because it may have an independent proof and is used later in the paper.