Let $M$ be a $n\times n$ triangular matrix, that entries are $0$ and $1$ , and such that diagonal entries are $1$. A row or a column will be said to be small, if its numer of $1$ is at most $(n+1)/2$. A row will be said 2-irreducible if it is not the conjonction of exactly two other rows. (The row $(a_1,a_2...,a_n)$ is the conjonction of the rows $(b_1,...,b_n) $ and $(c_1,...,c_n)$ iff $a_i=b_i.c_i$ for all non zero positive integer $i\leq n$).
Let's now suppose that every column is small in our matrix $M$
is there a row of $M$ that is small and 2-irreducible?
Note that if we ask $M$ to be the adjacency matrix of a lattice, then the question is equivalent to the Frankl conjecture. Note also that if one can broke the new conjecture, there is still an intermediary question by asking $M$ to be the adjacency matrix of a (finite) partial order. In a lattice $2$-irreducible and irreducible is the same notion, but in a general partial order, not being the upper bound of two distinct members is different from not being the upper bound of some subset. Indeed if we ask the same thing for "irreducible" but not 2- irreducible, one can easily build a counterexample with $M$ adjacency matrix of some partial order