Matrix form of Frankl's union-closed sets conjecture

Let $M=(m_{ij})$ be a non-zero binary matrix of size $n\times m$ with distinct rows. For binary rows, write $M_{i-}\lor M_{j-}$ for their coordinatewise logical disjunction, and denote the columns of $M$ by $M_{-k}$. Matrix form of Frankl's conjecture. If for every pair of rows $M_{i-}$ and $M_{j-}$ the row $M_{i-}\lor M_{j-}$ also occurs in $M$, then there exists a column $M_{-k}$ containing at least $n/2$ ones.

This is a binary-matrix reformulation of the union-closed sets conjecture, expressing union closure as closure of the rows under coordinatewise disjunction. Its general status is open, alongside the original set-system formulation.