This question was asked several months ago on Math.SE, but remains unsolved.
For any collection of permutations of $\{1,2,\dots,n\}$, we say that it realizes a directed multigraph with $1,2,\dots,n$ as vertices, such that there is an edge from $i$ to $j$ if $i$ appears before $j$ in at least half of the permutations, and there is an extra edge from $i$ to $j$ (so, two edges in total) if $i$ appears before $j$ in all of the permutations.
Is it true that for any collection of permutations that realizes a multigraph $G$, there exists a collection of two permutations (not necessarily distinct, and not necessarily from the original collection) that realizes a multigraph $H$ such that the edge set of $H$ is a superset of the edge set of $G$?
For example, if we have a collection of permutations
$(1,2,3,4), (1,4,3,2), (2,3,1,4), (3,1,2,4)$
the graph $G$ contains two edges $(1,4)$ and one edge $(1,2), (1,3), (2,3), (2,4), (3,1), (3,2), (3,4)$ each. Then we can choose the collection
$(1,2,3,4), (3,1,2,4)$
to fulfill the condition, since the realized graph $H$ contains two edges $(1,2), (1,4), (2,4), (3,4)$ each and one edge $(1,3), (2,3), (3,1), (3,2)$ each.
