The complete-bipartite super-graceful classification conjecture

Let $K(m,n)$ be the complete bipartite graph with part sizes $m$ and $n$, and let $k$-super graceful have the labeling meaning used in the paper. Complete-bipartite classification conjecture. If $m=1$ and $n$ is prime, or if $n\geq m\geq 2$, then

$$K(m,n)\text{ is $k$-super graceful if and only if }k\in\{1,m,n\}.$$

This would classify the permitted parameters for the stated complete bipartite graphs; the source provides partial results but does not resolve the conjecture.