The complete-bipartite majority out-domination conjecture

Let $K_{r,s}$ be a complete bipartite graph with $2\leq r\leq s$, and let $DOM^+_{maj}(K_{r,s})$ denote the minimum cardinality of a set that is a majority out-dominating set in an orientation of $K_{r,s}$, as defined in the source.

Complete-bipartite majority out-domination conjecture.

$$DOM^+_{maj}(K_{r,s})=\begin{cases}2&\text{if }r+s\text{ is even},\\\\3&\text{if }r+s\text{ is odd}. \end{cases}$$

The conjecture follows a proposition establishing the corresponding signed-function value $dom^+_{maj}(K_{r,s})=4-(r+s)$ for every orientation. The stronger cardinality assertion for $DOM^+_{maj}$ is stated here without a resolution.