The 1089 graph divisibility conjecture for reverse multiples

Let $g$ and $k$ be parameters of a $(g,k)$-reverse multiple and let its associated Young graph be the $(g,k)$ Young graph. The 1089 graph is the Young graph isomorphic to the $(10,4)$ and $(10,9)$ Young graphs.

1089 graph conjecture. The $(g,k)$ Young graph is isomorphic to the 1089 graph if and only if $k+1$ divides $g$.

The conjecture would explain the many occurrences of the 1089 graph in the computed examples; it is known when $k=g-1$, but the general case remains open.