Latent subsets for dual intersecting systems

A collection F of subsets of $\{ 1,2, \cdots ,n\} $ is a dual intersecting system if no two sets in F have an empty intersection or an exhaustive union. Let $f_j $ be the number of sets in F that contain point j, and let $f_j^L $ be the number of sets not in F but included in some set in F that contain j. We conjecture that if F is a dual intersecting system, then $f_j^L \geqq f_j $ for some j in $\{ 1, \cdots ,n\} $. This is shown to be true if either min $f_j \leqq n$ or min $f_j < 8$.