# Opposite vertices of base polyhedra

Is it true that if all vertices of a base polyhedron B are in $\{0,1,-1\}^n$ and $0 \in B$, then B has a vertex v such that -v is also a vertex?

## Remarks

This would imply the following conjecture of Frank:

Conjecture. If P is an integer 2-polymatroid which contains the 1 (all-ones) vector, then it has a vertex v such that 2-v is in P.

It is known by the Matroid union theorem that if the all-1/2 vector is in a matroid polyhedron, then the ground set of the matroid can be covered by 2 bases. This implies that the above conjecture is true if every vertex of P has only even coordinates.

A possible strengthening is the following.

Question. Is it true that if all vertices of an integer g-polymatroid Q are in $\{0,1,-1\}^n$ and $0 \in Q$, then Q has a vertex v such that -v is in Q?