Strong union-closed sets conjecture for matrices

Let $F$ be a reduced union-closed matrix, let $F^T$ be its transpose, and let $\overline{F}$ be the characteristic matrix of the complements of the represented sets. Write $\Sigma X$ for the sum of all entries of $X$. Matrix difference formulation of the strong union-closed sets conjecture.

$$\Sigma[FF^T] \ge \Sigma[F\overline{F}^T],$$ $$\Sigma[FF^T] \ge \Sigma[\overline{F}F^T].$$

The source obtains this formulation by translating the difference form of the strong union-closed sets conjecture into characteristic-matrix notation.