Complete bipartite graph monochromatic $k$-connection conjecture

Let $K_{s,t}$ be the complete bipartite graph with parts of sizes $s$ and $t$. Complete bipartite monochromatic $k$-connection conjecture. For $t\ge s\ge k\ge 2$,

$$mc_k(K_{s,t})=st-kt+h_k(K_{s,t}),$$

and, for $t\ge k\ge 2$,

$$mc_k(K_{t,t})=t^2-kt+1.$$

The first assertion specializes the bipartite conjecture to complete bipartite graphs; the second uses that a minimum spanning $k$-connected subgraph of $K_{t,t}$ has $kt$ edges and that its corresponding $h_k$ value is $1$.