Question
Let $A$ be a real square matrix $[a_{ij}]$ and let $D(A)=\sum_{i\geq j}a_{ij}$ . Find the maximum of $D(A)$ for the group of orthogonal matrices of given order $n$ ; and show that , as $n\to\infty$ , the maximum is asymptotic to $$\frac{n\log n}{\pi}$$
My attempt
For $n=3$, with a PC, I find a maximum $M$ reached for$$A\approx\begin{pmatrix}0.871119& -0.483435& -0.0862679\\0.387684&0.784851& -0.483435\\ 0.301417&0.387684& 0.871119\end{pmatrix}$$Note that the considered $(a_{i,j})$ are $>0$ and the pseudo-symmetry of $A$.
We can notice that the maximum exist by some compacity and continuity argument, and try to construct a maximizing sequences. The idea I tried but I am not able to complete the proof.
Edit-1
For $n=2$ it's easy. We find a maximum of $\sqrt{5}$ reached for$$A=\begin{pmatrix}\dfrac{2}{\sqrt{5}}&-\dfrac{1}{\sqrt{5}}\\\dfrac{1}{\sqrt{5}}&\dfrac{2}{\sqrt{5}}\end{pmatrix}$$
Edit-2
With $18$ significant digits, $M\approx 3.60387547160967650$.It seems that $M$ is a root of $x^3-2x^2-8x+8$ (?).