I am looking for the largest Wasserstein distance to the uniform distribution among all probability distributions with uniform marginals.

More specifically, let $\Xi=\{1,2,\ldots,N\}^2$, and let $\nu$ be a uniform distribution on $\Xi$, namely, $\nu_{ij}:=\nu(\{(i,j)\})=\frac{1}{N^2}$, for all $1\leq i,j\leq N$. Consider the problem \begin{equation} \max_{\mu\geq0} \bigg \{ W(\mu,\nu):\ \sum_{i}\mu_{ij} = \frac{1}{N}, \forall j,\ \sum_{j} \mu_{ij}=\frac{1}{N}, \forall i \bigg\}, \tag{1} \end{equation} where $W_1(\mu,\nu)$ is the Wasserstein distance between probability distribution $\mu:=\{\mu_{ij}\}_{1\leq i,j\leq N}$ and $\nu$, defined as $$ W_1(\mu,\nu) := \min_{\pi\geq0} \bigg\{\sum_{1\leq i,j,i',j' \leq N} ||(i,j)-(i',j')||_1 \pi_{(i,j),(i',j')}:\ \sum_{i,j} \pi_{(i,j),(i',j')} = \nu_{i'j'},\forall i',j',\ \sum_{i',j'} \pi_{(i,j),(i',j')} = \mu_{ij},\ \forall i,j\bigg\}. $$

My conjecture is that the maximizer of problem $(1)$ is given by $\mu_{ij}=\frac{1}{N}{1}_{\{i=j\}}$, or $\mu_{ij}=\frac{1}{N}{1}_{\{i+j=N+1\}}$, namely, the comonotonic/countermonotonic distribution. But how to prove/disprove it? Also, if it is true, could the result be extended to the multivariate case, namely, $\Xi=\{1,2,\ldots,N\}^K$ for $K>2$, or be extended for norms other than $\ell_1$-norm, or other $W_p$ distance ($p>1$)?

Update: Steve provides an affirmative answer for $\ell_1$-norm with $W_1$ distance in the case of $K=2$, and I provide a proof for $\ell_2$ norm with $W_2$ distance for all $K\geq2$. I am wondering if we can get some other result, such as $\ell_1$-norm with $K>2$, or $\ell_\infty$-norm.

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