Let $A$ be 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\begin{bmatrix}
\cos\phi \cos\theta \cos\psi - \sin\phi \sin\psi & 
\sin\phi \cos\theta \cos\psi + \cos\phi \sin\psi & 
-\sin\theta \cos\psi                                   \\
-\cos\phi \cos\theta \sin\psi - \sin\phi \cos\psi & 
-\sin\phi \cos\theta \sin\psi + \cos\phi \cos\psi & 
\sin\theta \sin\psi                                    \\
\cos\phi \sin\theta & \sin\phi \sin\theta & \cos\theta
\end{bmatrix}
\end{equation}




Let $B=QAQ^\top$ and 
\begin{equation}
F=\frac14\left|B_{12}^2-B_{21}^2\right|+\frac14\left|B_{13}^2-B_{31}^2\right|+\frac14\left|B_{23}^2-B_{23}^2\right|
\end{equation} 

Is is possible to prove the concavity of $F$ for a given $A$? or otherwise find an example of $A$ for which $F$ is not concave ?

I can find numerically the maximum of $F$ for a set of random matrix $A$, but I am enable to find a rigorous way that this maximum is indeed a global or local one.