I confirmed via integer linear programming that the maximum value is $2k(n-k)$ for $n \le 20$ and $k>n/2$.  The formulation is the same as in [my answer to the linked question][1], except for the following changes:
+ The node set is $N = \{1,\dots,2n\}$.
+ The candidate edge set is $E = \{i \in N, j \in N: i \le n < j\}$.
+ Constraints $(3)$ and $(4)$ become
\begin{align}
k - (d_i - d_j) &\le (k + n) (1 - u_{i,j}) &&\text{for $(i,j)\in E$} \tag3\\
k - (d_j - d_i) &\le (k + n) (1 - v_{i,j}) &&\text{for $(i,j)\in E$} \tag4\\
\end{align}


  [1]: https://mathoverflow.net/a/389768/141766