Denote $ C=OPQ^T $, where $O,P,Q$ are all real matrices with sizes $ D^2 \times n $, $ n \times n $ and $ d^2 \times n $ respectively. Assume $ d < D $.
Moreover, $P$ is diagonal, and the sum of its entries is $1$; the first rows of $O,Q$ are all $1$-s; all the columns of $O$ have norm $ \sqrt{D} $, and all the columns of $Q$ have norm $ \sqrt{d} $.
Given these constraints, how can one find an upper bound for the product of all the singular values of $C$ - i.e., $ \sqrt{\det \left( C^T C \right)} $?
Edit: I seek an upper bound independent of $n$. I.e., we may freely choose the value of $n$ yielding the maximal determinant.
Moreover, I have a hypothesis for the tight upper bound: I think it is obtained for $ n=d^2 $, $ P = \frac{1}{d^2} 1 $ ($P$ is a scalar matrix), and $O,Q$ are matrices whose columns form $d^2$ equiangular lines in $\mathbb{R}^{D^2}, \mathbb{R}^{d^2}$ respectively.
For example, for $d=2,D=3$, I believe the upper bound is obtained for $n=d^2$ and the following matrices: \begin{equation} O = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & -\frac{4\sqrt{3}}{9} & \frac{2\sqrt{3}}{9} & \frac{2\sqrt{3}}{9} \\ 0 & 0 & 0 & 0 \\ \sqrt{\frac{3}{2}} & -\frac{7\sqrt{6}}{18} & -\frac{\sqrt{6}}{18} & -\frac{\sqrt{6}}{18} \\ 0 & 0 & -\frac{2}{3} & \frac{2}{3} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{2\sqrt{2}}{3} & \frac{2\sqrt{2}}{3} \\ 0 & 0 & 0 & 0 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{bmatrix} \end{equation} and \begin{equation} P = \begin{bmatrix} \frac{1}{4} & 0 & 0 & 0 \\ 0 & \frac{1}{4} & 0 & 0 \\ 0 & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & \frac{1}{4} \end{bmatrix}, Q = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -\frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} \\ 0 & \frac{2\sqrt{2}}{3} & -\frac{\sqrt{2}}{3} & -\frac{\sqrt{2}}{3} \\ 0 & 0 & \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} \end{bmatrix} . \end{equation} Note the relation to tetrahedrons. Also, dividing the first row of $Q$ by $ \sqrt{d+1} = \sqrt{3} $ makes it a scalar times orthogonal, where the scalar is $ \frac{d}{\sqrt{d+1}} $.
I am not certain about the validity of this hypothesis, and even less certain about how to prove it.