Frankl's conjecture for reduced intersection-closed matrices

Let $F \in \mathbb{Q}^{m \times n}$ be a reduced intersection-closed matrix, and let $e_k$ be the $k$-th unit coordinate vector. Write $\Sigma X$ for the sum of all entries of a matrix or vector. Frankl's intersection-matrix formulation.

$$2 \min\{\Sigma [F e_k]: k=1,\ldots,n\} \le m.$$

This is an equivalent matrix formulation of Frankl's conjecture. The source states the stronger version that the inequality is strict unless $m=2^n$.