Among all the probability matrices

\begin{equation*}
P =
\left(\begin{array}{cccc}
p_{00} & p_{01} & \ldots & p_{0,J-1} \\
p_{10} & p_{11} & \ldots & p_{1,J-1} \\
\vdots & \vdots & \ddots &  \vdots \\
p_{J-1,0} & p_{J-1, 1} & \ldots & p_{J-1,J-1} \\
\end{array} \right)
\quad
(0 \le p_{kl} \le 1; \; k,l = 0, \ldots, J-1)
\end{equation*}
with fixed row and column sums
\begin{equation*}
p_{k+} = \sum_{l^\prime=0}^{J-1}p_{kl^\prime},
\quad
p_{+l} = \sum_{k^\prime =0}^{J-1}p_{k^\prime l}
\quad
(k, l = 0, 1, \ldots, J-1),
\end{equation*}
Calculate the closed-form expressions for the minimum and maximum values of
\begin{equation*}
\Delta = \mathop{\sum\sum}_{ k > l}p_{kl}  - \mathop{\sum\sum}_{ k < l}p_{kl}.
\end{equation*}


Some background of this question:

This is a active research problem I am currently pursuing. I can't prove that it indeed has a closed form solution, but I tend to believe so. I have tried very hard and therefore I want to solicit any hints or point of directions. On a related note,  I have obtained closed form solutions of very similar objective functions, another reason why I believe the solution exists for this one as well. For more details see dash.harvard.edu/handle/1/23845443